However, if we move the charge +𝑞 very close to one of the four faces, the electric field becomes asymmetrical so I cannot apply Gauss law directly.
That's correct, if you only want the electric flux through each face individually and not the total flux over all the faces. To determine the flux on each face $A$, you would need to integrate the flux over that face, or
$$Φ_{A}=\int_A \overrightarrow E.d\overrightarrow A$$
Since the flux over each surface is non-uniform, the calculation would obviously be difficult.
Hope this helps.