Gauss' law states that the net flux across a closed surface equals the net charge enclosed divided by the electrical permittivity of the space. Moving the charge around within the enclosed volume does not change the net flux crossing the surface and Gauss' law still applies. Moving the charge just changes the distribution of the flux across the different portions of the surface, not the net flux over the entire surface.

Hope this helps.

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.