Instead of separating the two pieces, it seems highly favourable to drill small holes through the black plastic layer, allowing you to remove the 4 screws without separation

Accoustically, this probably isn't a great idea. So, afterwards, you would want to shake out as much of the plastic shavings as possible, and plug the holes as flush as possible using hot glue: push a drop of hot glue in the area with a finger or tool that has been wet with dish soap and water. It will prevent the glue from sticking on release.

In the picture, I've done my best to mark off approximate locations to be drilled. If off, you could either make larger holes or a separate hole, depending on your preference. Be very careful - take your time and drill with light/no pressure!

Normally, I don't like making permanent modifications when possible. In this case, it seems almost unavoidable