Again, not exactly the best idea, but with a small enough screw, very carefully drilled tiny hole, and some reinforcing glue, you might have another option there.
There already IS a hole in the bottom "half", but I’m not sure having a screw head or a nut here wouldn’t mess with the membrane under it, nor how the button is sitting. Might still try it if I don’t find any other solutions.
French peoples are too.