The screws have arrived, and they are good. The 4mm long ones are just the right size to go into the middle layer, and the 5mm ones are perfect for holding both layers. Of course now I am in the process of rethinking that front layer anyways, but if I keep using screws, I know where to get them.

From the left, the original screw I used, the same kind with its tip cut off to make it not stick out of the case, the new 5mm screws, and the new 4mm screw.