Good news everyone ! I found the solution a workaround.
So the problem is, Godot (Or maybe my graphic card, I really don't know) is unable to process transparency correctly when the light comes from the front. But this only true on half-transparent pixels (those which are partially opaque).
The grids I use in my example are not completely opaque. Forcing them to have binary transparency solved the issue.
The workaround consists in two steps:
1) Make sure your source image only has binary transparency (each pixel is either 100% transparent or 100% opaque). You can use GIMP's "alpha threshold" option for this.
2) Disable mipmaps in the import tab. I know it's recommended for 3D, and disabling it makes the game both less optimized and slightly uglier, but if your material is far away, mipmaps is going to calculate semi-transparent pixels even though the source image only has binary transparency.
I hope this helps some people ! And if you have a better solution, please share it
Thanks to Megalomaniac for their time ! I greatly appreciated it !