You start approaching the problem by the right way but at an intermediate step what you missed was to express the electric field $\,\mathbf E\,$ in terms of the potentials
\begin{equation}
A^0\boldsymbol=\phi\,, \quad A^1\boldsymbol=A_{\rm x}\,, \quad A^2\boldsymbol=A_{\rm y}\,, \quad A^3\boldsymbol=A_{\rm z}
\tag{01}\label{01}
\end{equation}
since these are the $''$general coordinates$''$.
The problem is if this Lorentz invariant scalar of the electromagnetic field
\begin{equation}
\boldsymbol-\mathbf E\boldsymbol\cdot \mathbf B
\tag{02}\label{02}
\end{equation}
could be expressed as the 4-divergence of a 4-dimensional vector so its addition to the well-known Lagrangian density doesn't affect the Equations Of Motion, that is the Maxwell equations.
Expressing the fields in terms of the potentials
\begin{equation}
\boldsymbol-\mathbf E\boldsymbol\cdot \mathbf B\boldsymbol=\left(\boldsymbol\nabla\phi\boldsymbol+\dfrac{\partial\mathbf A }{\partial t}\right)\boldsymbol\cdot \left(\boldsymbol{\nabla\times}\mathbf A\right)
\tag{03}\label{03}
\end{equation}
our target would be to check if it's possible to express it as follows
\begin{equation}
\boldsymbol-\mathbf E\boldsymbol\cdot \mathbf B\boldsymbol=\left(\boldsymbol\nabla\phi\boldsymbol+\dfrac{\partial\mathbf A }{\partial t}\right)\boldsymbol\cdot \left(\boldsymbol{\nabla\times}\mathbf A\right)\boldsymbol=\dfrac{\partial h^0 }{\partial t}\boldsymbol+\boldsymbol{\nabla\cdot}\mathbf h
\tag{04}\label{04}
\end{equation}
where $\,h^0\,$ and $\,\mathbf h\,$ scalar and 3-vector functions of the potentials respectively.
Now, we could verify that the following 4-dimensional vector
\begin{equation}
\begin{split}
h^0 & \boldsymbol= \tfrac{1}{2}\mathbf A\boldsymbol\cdot\left(\boldsymbol{\nabla\times}\mathbf A\right)\\
\mathbf h & \boldsymbol=\phi\left(\boldsymbol{\nabla\times}\mathbf A\right)\boldsymbol+\tfrac{1}{2}\left(\mathbf A\boldsymbol\times\dfrac{\partial\mathbf A }{\partial t}\right)
\end{split}
\tag{05}\label{05}
\end{equation}
satisfies equation \eqref{04}.
The following vector formulas would be useful
\begin{equation}
\begin{split}
\boldsymbol{\nabla\cdot}\left(\psi\,\mathbf a\right) & \boldsymbol= \mathbf a\boldsymbol\cdot\boldsymbol\nabla \psi\boldsymbol+\psi\boldsymbol{\nabla\cdot}\mathbf a\\
\boldsymbol{\nabla\cdot}\left(\mathbf a\boldsymbol\times\mathbf b\right) & \boldsymbol= \mathbf b\boldsymbol\cdot\left(\boldsymbol{\nabla\times}\mathbf a\right)\boldsymbol-\mathbf a\boldsymbol\cdot\left(\boldsymbol{\nabla\times}\mathbf b\right)\\
\end{split}
\tag{06}\label{06}
\end{equation}
$=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!$
ADDENDUM A :
$\texttt{ Derivation of the 4-dimensional vector }$ \eqref{05} $\texttt{ that satisfies equation }$ \eqref{04}
From equation \eqref{03}
\begin{equation}
\boldsymbol-\mathbf E\boldsymbol\cdot \mathbf B\boldsymbol=\underbrace{\boldsymbol\nabla\phi\boldsymbol\cdot \left(\boldsymbol{\nabla\times}\mathbf A\right)\vphantom{\dfrac{\partial\mathbf A }{\partial t}}}_{\boxed{\,1\,}}\boldsymbol+\underbrace{\dfrac{\partial\mathbf A }{\partial t}\boldsymbol\cdot \left(\boldsymbol{\nabla\times}\mathbf A\right)}_{\boxed{\,2\,}}
\tag{A-01}\label{A-01}
\end{equation}
Because of the first vector formula \eqref{06} and due to the fact that $\,\boldsymbol{\nabla\cdot}\left(\boldsymbol{\nabla\times}\mathbf A\right)\boldsymbol=0\,$ we have
\begin{equation}
\boxed{1\,}\boldsymbol=\boldsymbol\nabla\phi\boldsymbol\cdot \left(\boldsymbol{\nabla\times}\mathbf A\right)\vphantom{\dfrac{\partial\mathbf A }{\partial t}}\boldsymbol=\boldsymbol\nabla\boldsymbol\cdot\bigl[\phi \left(\boldsymbol{\nabla\times}\mathbf A\right)\bigr]
\tag{A-02}\label{A-02}
\end{equation}
so expressing the term $\,\boxed{1\,}\,$ as the divergence of a 3-vector function of the potentials.
The second vector formula \eqref{06} with $\,\mathbf a \boldsymbol=\mathbf A\,$ and $\,\mathbf b \boldsymbol=\partial\mathbf A /\partial t\,$ yields
\begin{equation}
\boxed{2\,}\boldsymbol=\dfrac{\partial\mathbf A }{\partial t}\boldsymbol\cdot \left(\boldsymbol{\nabla\times}\mathbf A\right)\boldsymbol=\mathbf A\boldsymbol\cdot \left(\boldsymbol{\nabla\times}\dfrac{\partial\mathbf A }{\partial t}\right)\boldsymbol+\boldsymbol{\nabla\cdot} \left(\mathbf A\boldsymbol\times\dfrac{\partial\mathbf A }{\partial t}\right)
\tag{A-03}\label{A-03}
\end{equation}
But from the differentiation of a product
\begin{equation}
\boxed{2\,}\boldsymbol=\dfrac{\partial\mathbf A }{\partial t}\boldsymbol\cdot \left(\boldsymbol{\nabla\times}\mathbf A\right)\boldsymbol=\dfrac{\partial}{\partial t}\left[\mathbf A\boldsymbol\cdot\left(\boldsymbol{\nabla\times}\mathbf A\right)\vphantom{\dfrac{\partial\mathbf A }{\partial t}}\right]\boldsymbol-\mathbf A\boldsymbol\cdot \left(\boldsymbol{\nabla\times}\dfrac{\partial\mathbf A }{\partial t}\right)
\tag{A-04}\label{A-04}
\end{equation}
Adding equations \eqref{A-03} and \eqref{A-04} side by side yields
\begin{equation}
2\cdot\boxed{2\,}\boldsymbol=2\dfrac{\partial\mathbf A }{\partial t}\boldsymbol\cdot \left(\boldsymbol{\nabla\times}\mathbf A\right)\boldsymbol=\dfrac{\partial}{\partial t}\left[\mathbf A\boldsymbol\cdot\left(\boldsymbol{\nabla\times}\mathbf A\right)\vphantom{\dfrac{\partial\mathbf A }{\partial t}}\right]\boldsymbol+\boldsymbol{\nabla\cdot} \left(\mathbf A\boldsymbol\times\dfrac{\partial\mathbf A }{\partial t}\right)
\tag{A-05}\label{A-05}
\end{equation}
so
\begin{equation}
\boxed{2\,}\boldsymbol=\dfrac{\partial\mathbf A }{\partial t}\boldsymbol\cdot \left(\boldsymbol{\nabla\times}\mathbf A\right)\boldsymbol=\dfrac{\partial}{\partial t}\left[\tfrac{1}{2}\mathbf A\boldsymbol\cdot\left(\boldsymbol{\nabla\times}\mathbf A\right)\vphantom{\dfrac{\partial\mathbf A }{\partial t}}\right]\boldsymbol+\boldsymbol{\nabla\cdot} \left[\tfrac{1}{2}\left(\mathbf A\boldsymbol\times\dfrac{\partial\mathbf A }{\partial t}\right)\right]
\tag{A-06}\label{A-06}
\end{equation}
Finally adding equations \eqref{A-02} and \eqref{A-06} we have
\begin{equation}
\begin{split}
\boldsymbol-\mathbf E\boldsymbol\cdot \mathbf B & \boldsymbol=\boxed{1\,}\boldsymbol+\boxed{2\,}\\
& \boldsymbol=\dfrac{\partial}{\partial t}\underbrace{\left[\tfrac{1}{2}\mathbf A\boldsymbol\cdot\left(\boldsymbol{\nabla\times}\mathbf A\right)\vphantom{\dfrac{\partial\mathbf A }{\partial t}}\right]}_{h^0}\boldsymbol+\boldsymbol{\nabla\cdot} \underbrace{\left[\phi\left(\boldsymbol{\nabla\times}\mathbf A\right)\boldsymbol+\tfrac{1}{2}\left(\mathbf A\boldsymbol\times\dfrac{\partial\mathbf A }{\partial t}\right)\right]}_{\mathbf h} \\
\end{split}
\tag{A-07}\label{A-07}
\end{equation}
qed.
$=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!$
ADDENDUM B :
$\texttt{ The unaffected Euler-Lagrange (Maxwell's) Equations}$
That the addition of the term $\,\boldsymbol-\mathbf E\boldsymbol\cdot \mathbf B\,$ doesn't affect the Equations Of Motion, that is the Maxwell equations, could be proved directly in these Euler-Lagrange Equations.
So let the contravariant space-time position 4-vector, the contravariant potential 4-vector and the covariant 4-gradient
\begin{equation}
\begin{split}
x^0 & \boldsymbol=t\,, \quad x^1\boldsymbol=\mathrm x\,, \quad x^2\boldsymbol=\mathrm y\,, \quad x^3\boldsymbol=\mathrm z\\
A^0 & \boldsymbol=\phi\,, \quad A^1\boldsymbol=A_{\rm x}\,, \quad A^2\boldsymbol=A_{\rm y}\,, \quad A^3\boldsymbol=A_{\rm z}\\
\partial_k & \boldsymbol\equiv\dfrac{\partial}{\partial x^k}\,, \qquad k=0,1,2,3
\end{split}
\tag{B-01}\label{B-01}
\end{equation}
Mixing tensor calculus and 3-vector notations consider the Lagrangian
\begin{equation}
\mathcal L'\left(A^\mu, \partial_k A^\mu\right) \boldsymbol=\boldsymbol-\mathbf E\boldsymbol\cdot \mathbf B\boldsymbol=\left(\boldsymbol\nabla\phi\boldsymbol+\dfrac{\partial\mathbf A }{\partial t}\right)\boldsymbol\cdot \left(\boldsymbol{\nabla\times}\mathbf A\right)
\tag{B-02}\label{B-02}
\end{equation}
Using the summation convention for $k\boldsymbol=0,1,2,3$ we have the following 4 scalar Euler-Lagrange Equations
\begin{equation}
\begin{split}
&\partial_k\left[\dfrac{\partial \mathcal L' }{\partial \left( \partial_k A^\mu\right)}\right]\boldsymbol- \frac{\partial \mathcal L'}{\partial A^\mu}\boldsymbol=0\:, \quad \mu\boldsymbol=0,1,2,3 \\
&^{\left(k\boldsymbol=0,1,2,3\right)}\\
\end{split}
\tag{B-03}\label{B-03}
\end{equation}
Separating the time from the space partial derivatives and using the summation convention for the space indices $\nu\boldsymbol=1,2,3$ above equations are expressed more explicitly as follows
\begin{equation}
\begin{split}
\frac{\partial }{\partial t}\left[\frac{\partial \mathcal L'}{\partial \left(\dfrac{\partial A^\mu}{\partial t}\right)}\right]\boldsymbol+&\frac{\partial }{\partial x^\nu}\left[\frac{\partial \mathcal L'}{\partial \left(\dfrac{\partial A^\mu}{\partial x^\nu}\right)}\right]\boldsymbol- \frac{\partial \mathcal L'}{\partial A^\mu}\boldsymbol=0\:, \quad \mu\boldsymbol=0,1,2,3\\
&^{\left(\nu\boldsymbol=1,2,3\right)}\\
\end{split}
\tag{B-04}\label{B-04}
\end{equation}
For $\mu\boldsymbol=0$, that is for the equation with respect to the scalar potential $\,A^0\boldsymbol=\phi$
\begin{equation}
\begin{split}
\frac{\partial }{\partial t}\left[\frac{\partial \mathcal L'}{\partial \left(\dfrac{\partial A^0}{\partial t}\right)}\right]\boldsymbol+&\frac{\partial }{\partial x^\nu}\left[\frac{\partial \mathcal L'}{\partial \left(\dfrac{\partial A^0}{\partial x^\nu}\right)}\right]- \frac{\partial \mathcal L'}{\partial A^0}\boldsymbol=0\\
&^{\left(\nu\boldsymbol=1,2,3\right)}\\
\end{split}
\tag{B-05}\label{B-05}
\end{equation}
or
\begin{equation}
\frac{\partial }{\partial t}\underbrace{\left[\frac{\partial \mathcal L'}{\partial \left(\dfrac{\partial \phi}{\partial t}\right)}\right]}_{0}\boldsymbol+\underbrace{\boldsymbol\nabla\boldsymbol\cdot\overbrace{\left[\frac{\partial \mathcal L'}{\partial \left(\boldsymbol\nabla\phi\right)}\right]}^{\boldsymbol{\nabla\times}\mathbf A}\vphantom{\frac{\partial \mathcal L'}{\partial \left(\dfrac{\partial \phi}{\partial t}\right)}}}_{0}\boldsymbol-\underbrace{\frac{\partial \mathcal L'}{\partial \phi}\vphantom{\frac{\partial \mathcal L'}{\partial \left(\dfrac{\partial \phi}{\partial t}\right)}}}_{0}\boldsymbol=0
\tag{B-06}\label{B-06}
\end{equation}
that is a null equation $0\boldsymbol=0$.
The same holds also for the 3 equations with respect to the components of the vector potential $\,\left(A^1,A^2,A^3\right)\boldsymbol=\mathbf A$
\begin{equation}
\underbrace{\frac{\partial }{\partial t}\overbrace{\left[\frac{\partial \mathcal L'}{\partial \left(\dfrac{\partial A^\mu}{\partial t}\right)}\right]}^{\left(\boldsymbol{\nabla\times}\mathbf A\right)^\mu}\boldsymbol+\overbrace{\frac{\partial }{\partial x^\nu}\left[\frac{\partial \mathcal L'}{\partial \left(\dfrac{\partial A^\mu}{\partial x_\nu}\right)}\right]}^{\boldsymbol-\tfrac{\partial \left(\boldsymbol{\nabla\times}\mathbf A\right)^\mu}{\partial t}}}_{0}\boldsymbol-\underbrace{\frac{\partial \mathcal L'}{\partial A^\mu}\vphantom{\frac{\partial \mathcal L'}{\partial \left(\dfrac{\partial A^\mu}{\partial t}\right)}}}_{0}\boldsymbol=0\:, \quad \mu\boldsymbol=1,2,3
\tag{B-07}\label{B-07}
\end{equation}
where $\,\left(\boldsymbol{\nabla\times}\mathbf A\right)^\mu\,$ the $\mu\boldsymbol-$ component of the 3-vector $\,\left(\boldsymbol{\nabla\times}\mathbf A\right)$.
In above equation the proof that
\begin{equation}
\begin{split}
&\frac{\partial }{\partial x^\nu}\left[\frac{\partial \mathcal L'}{\partial \left(\dfrac{\partial A^\mu}{\partial x^\nu}\right)}\right]\boldsymbol=\boldsymbol-\dfrac{\partial \left(\boldsymbol{\nabla\times}\mathbf A\right)^\mu}{\partial t}\\
&^{\left(\nu\boldsymbol=1,2,3\right)}\\
\end{split}
\tag{B-08}\label{B-08}
\end{equation}
would be clear if \eqref{B-02} is expressed with all details
\begin{equation}
\begin{split}
\mathcal L'\left(A^\mu, \partial_k A^\mu\right)& \boldsymbol=\boldsymbol-\mathbf E\boldsymbol\cdot \mathbf B\boldsymbol=\left(\boldsymbol\nabla\phi\boldsymbol+\dfrac{\partial\mathbf A }{\partial t}\right)\boldsymbol\cdot \left(\boldsymbol{\nabla\times}\mathbf A\right)\\
& \boldsymbol=\left(\dfrac{\partial\phi }{\partial x^1}\boldsymbol+\dfrac{\partial A^1}{\partial t}\right)\left(\dfrac{\partial A^3 }{\partial x^2}\boldsymbol-\dfrac{\partial A^2}{\partial x^3}\right)\boldsymbol+\left(\dfrac{\partial\phi }{\partial x^2}\boldsymbol+\dfrac{\partial A^2}{\partial t}\right)\left(\dfrac{\partial A^1 }{\partial x^3}\boldsymbol-\dfrac{\partial A^3}{\partial x^1}\right)\\
& \hphantom{\boldsymbol=}\boldsymbol+\left(\dfrac{\partial\phi }{\partial x^3}\boldsymbol+\dfrac{\partial A^3}{\partial t}\right)\left(\dfrac{\partial A^2 }{\partial x^1}\boldsymbol-\dfrac{\partial A^1}{\partial x^2}\right)\\
\end{split}
\tag{B-09}\label{B-09}
\end{equation}
For example
\begin{equation}
\begin{split}
\frac{\partial }{\partial x^\nu}\left[\frac{\partial \mathcal L'}{\partial \left(\dfrac{\partial A^1}{\partial x^\nu}\right)}\right]&\boldsymbol=\frac{\partial }{\partial x^1}\overbrace{\left[\frac{\partial \mathcal L'}{\partial \left(\dfrac{\partial A^1}{\partial x^1}\right)}\right]}^{0}\boldsymbol+\frac{\partial }{\partial x^2}\overbrace{\left[\frac{\partial \mathcal L'}{\partial \left(\dfrac{\partial A^1}{\partial x^2}\right)}\right]}^{\boldsymbol-\left(\tfrac{\partial\phi }{\partial x^3}\boldsymbol+\tfrac{\partial A^3}{\partial t}\right)}\boldsymbol+\frac{\partial }{\partial x^3}\overbrace{\left[\frac{\partial \mathcal L'}{\partial \left(\dfrac{\partial A^1}{\partial x^3}\right)}\right]}^{\boldsymbol+\left(\tfrac{\partial\phi }{\partial x^2}\boldsymbol+\tfrac{\partial A^2}{\partial t}\right)}\\
& \boldsymbol=\boldsymbol-\frac{\partial }{\partial x^2}\left(\dfrac{\partial\phi }{\partial x^3}\boldsymbol+\dfrac{\partial A^3}{\partial t}\right)\boldsymbol+\frac{\partial }{\partial x^3}\left(\dfrac{\partial\phi }{\partial x^2}\boldsymbol+\dfrac{\partial A^2}{\partial t}\right)\\
& \boldsymbol=\boldsymbol-\frac{\partial }{\partial t}\left(\dfrac{\partial A^3}{\partial x^2}\boldsymbol-\dfrac{\partial A^2}{\partial x^3}\right)\boldsymbol=\boldsymbol-\dfrac{\partial \left(\boldsymbol{\nabla\times}\mathbf A\right)^1}{\partial t}\\
\end{split}
\tag{B-10}\label{B-10}
\end{equation}
$=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!$
ADDENDUM C :
$\texttt{ Is the 4-vector }\left(h^0,\mathbf h\right)\texttt{ of equation \eqref{05} a Lorentz 4-vector?}$
The 4-divergence of a Lorentz 4-vector function of space-time coordinates is a Lorentz invariant. The inverse is not valid in general. That is, if the 4-divergence of a 4-vector function of space-time coordinates is a Lorentz invariant this doesn't imply that the 4-vector function is a Lorentz one.
So it would be reasonable to ask if the 4-vector function of \eqref{05} is a
Lorentz one
\begin{equation}
\mathbf H\boldsymbol=
\begin{bmatrix}
h^0\vphantom{\dfrac{a}{b}}\\
\vphantom{\dfrac{a}{b}}\\
\mathbf h\vphantom{\dfrac{a}{b}}\\
\vphantom{\dfrac{a}{b}}
\end{bmatrix}\boldsymbol=
\begin{bmatrix}
\tfrac{1}{2}\mathbf A\boldsymbol\cdot\left(\boldsymbol{\nabla\times}\mathbf A\right)\vphantom{\dfrac{a}{b}}\\
\vphantom{\dfrac{a}{b}}\\
\phi\left(\boldsymbol{\nabla\times}\mathbf A\right)\boldsymbol+\tfrac{1}{2}\left(\mathbf A\boldsymbol\times\dfrac{\partial\mathbf A }{\partial t}\right)\vphantom{\dfrac{a}{b}}\\
\vphantom{\dfrac{a}{b}}
\end{bmatrix}
\tag{C-01}\label{C-01}
\end{equation}
I found that the above 4-vector IS NOT a Lorentz 4-vector, but I omit the proof as being out of the main subject of the post.
$=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!$
Related 1 : Deriving Lagrangian density for electromagnetic field.
Related 2 : Squaring the E&M (Maxwell) field strength tensor.