Note that the probability current in the presence of a EM field is given by
$$
\mathbf{j}=\frac{1}{2m}\left(\psi^{*}\mathbf{p}\psi-\psi\mathbf{p}\psi^{*}-2q\mathbf{A}\psi^{*}\psi\right)
$$
As you note a local phase shift
$$
\psi^{\prime}=e^{iq\chi(\mathbf{r},t)/\hbar}\psi
$$
leads to a gauge transformation of the vector potential
$$
\mathbf{A}^{\prime}=\mathbf{A}+\nabla\chi
$$
Substituting these into the expression for the probability current gives
$$\begin{multline}
\mathbf{j}^{\prime}=\frac{1}{2m}\left(e^{-iq\chi(\mathbf{r},t)/\hbar}\psi^{*}\mathbf{p}e^{iq\chi(\mathbf{r},t)/\hbar}\psi-e^{iq\chi(\mathbf{r},t)/\hbar}\psi\mathbf{p}e^{-iq\chi(\mathbf{r},t)/\hbar}\psi^{*}\right.\\\left.-2q\mathbf{A}e^{-iq\chi(\mathbf{r},t)/\hbar}\psi^{*}e^{iq\chi(\mathbf{r},t)/\hbar}\psi-2q\nabla\chi(\mathbf{r},t) e^{-iq\chi(\mathbf{r},t)/\hbar}\psi^{*}e^{iq\chi(\mathbf{r},t)/\hbar}\psi\right)
\end{multline}$$
Operating with $\mathbf{p}\rightarrow-i\hbar\nabla$ one obtains
$$\begin{multline}
\mathbf{j}^{\prime}=\frac{1}{2m}\left(\psi^{*}\left(-i\hbar\right)\nabla\psi+\psi^{*}\psi\frac{iq}{\hbar}(-i\hbar)\nabla\chi(\mathbf{r},t)-\psi\left(-i\hbar\right)\nabla\psi^{*}\right.\\\left.-\psi^{*}\psi\left(-\frac{iq}{\hbar}\right)(-i\hbar)\nabla\chi(\mathbf{r},t)-2q\mathbf{A}\psi^{*}\psi-2q\nabla\chi(\mathbf{r},t)\psi^{*}\psi\right)
\end{multline}$$
Sorting it out one obtains
$$
\mathbf{j}^{\prime}=\frac{1}{2m}\left(\psi^{*}\mathbf{p}\psi-\psi\mathbf{p}\psi^{*}-2q\mathbf{A}\psi^{*}\psi\right)=\mathbf{j}
$$
thus the probability current is gauge invariant.