@User mentioned in the comments that
$$\sum _{n=1}^{\infty } \frac{16^n}{n^3 \binom{2 n}{n}^2}=8\pi\text{G}-14 \zeta (3)\tag1$$
$$\small{\sum _{n=1}^{\infty } \frac{16^n}{n^4 \binom{2 n}{n}^2}=64 \pi \Im(\text{Li}_3(1+i))+64 \text{Li}_4\left(\frac{1}{2}\right)-233 \zeta(4)-40 \ln ^2(2)\zeta(2)+\frac{8}{3}\ln ^4(2)}\tag2$$
I was able to prove $(1)$ but had some difficulty proving $(2)$. Any idea?
I am going to show my proof of $(1)$ hoping it helps you prove $(2)$:
We showed in this question that
$$\sum_{n=1}^\infty\frac{4^ny^n}{n^2{2n\choose n}}=2\int_0^y \frac{\arcsin \sqrt{x}}{\sqrt{x}\sqrt{1-x}}dx$$
multiply both sides by $\frac{1}{y\sqrt{1-y}}$ then $\int_0^1$ with respect to $y$ and use $\int_0^1\frac{y^{n-1}}{\sqrt{1-y}}dy=\frac{4^n}{n{2n\choose n}}$ we obtain
$$\sum _{n=1}^{\infty } \frac{16^n}{n^3 \binom{2 n}{n}^2}=2\int_0^1\int_0^y \frac{\arcsin \sqrt{x}}{y\sqrt{x}\sqrt{1-x}\sqrt{1-y}}dxdy$$
$$=2\int_0^1\frac{\arcsin\sqrt{x}}{\sqrt{x}\sqrt{1-x}}\left(\int_x^1\frac{dy}{y\sqrt{1-y}}\right)dx$$
$$=2\int_0^1\frac{\arcsin\sqrt{x}}{\sqrt{x}\sqrt{1-x}}\left(2\ln(1+\sqrt{1-x})-\ln x\right)dx$$
$$\overset{\sqrt{x}=\sin \theta}{=}8\int_0^{\pi/2}x\ln(1+\cos x)dx-8\int_0^{\pi/2}x\ln(\sin x)dx$$
$$=8\int_0^{\pi/2}x\ln(2\cos^2\frac x2)dx-8\int_0^{\pi/2}x\ln(\sin x)dx$$
$$=32\int_0^{\pi/4}x\ln(2\cos^2x)dx-8\int_0^{\pi/2}x\ln(\sin x)dx$$
$$=32\underbrace{\int_0^{\pi/4}x\ln(2)dx}_{\frac3{16}\ln(2)\zeta(2)}+64\underbrace{\int_0^{\pi/4}x\ln(\cos x)dx}_{\frac{\pi}{8}\text{G}-\frac3{16}\ln(2)\zeta(2)-\frac{21}{128}\zeta(3)}-8\underbrace{\int_0^{\pi/2}x\ln(\sin x)dx}_{\frac7{16}\zeta(3)-\frac34\ln(2)\zeta(2)}$$
$$=8\pi\text{G}-14 \zeta (3)$$
The last two integrals follow from using the Fourier series of $\ln(\cos x)$ and $\ln(\sin x)$.
All approaches are appreciated. Thank you.
Addendum: Here is an easier way to prove $(1)$:
We have
$$\arcsin^2(x)=\frac12\sum_{n=1}^\infty\frac{(2x)^{2n}}{n^2{2n\choose n}}$$
or
$$\sum_{n=1}^\infty\frac{4^nx^n}{n^2{2n\choose n}}=2\arcsin^2(\sqrt{x})$$
Divide both sides by $x\sqrt{1-x}$ then $\int_0^1$ and use $\int_0^1\frac{x^{n-1}}{\sqrt{1-x}}dx=\frac{4^n}{n{2n\choose n}}$ we have
$$\sum_{n=1}^\infty\frac{16^n}{n^3{2n\choose n}^2}=2\int_0^1\frac{\arcsin^2(\sqrt{x})}{x\sqrt{1-x}}dx$$
$$\overset{\sqrt{x}=\sin x}{=}4\int_0^{\pi/2}x^2 \csc(x)dx$$
$$\overset{IBP}{=}-8\int_0^{\pi/4} x\ln(\tan\frac x2)dx=8\pi\text{G}-14\zeta(3)$$
where the last result follows from the Fourier series of $\ln(\tan\frac x2)$.