I started making the housing out of plywood and some scrap wood pieces.
Holes were drilled out of the front panel for the buttons: A, B, C, D, Coin and Start, as well as for the stick. Wood blocks on the side provide solidity and the desired housing thickness.
The contour of the PCB was chiseled out to give access to the USB-C port.
Another sheet of plywood was added to provde a bit of additional thickness, and to cover the four threaded inserts.
The top plywood panel was finally too thin to support the joystick with screws. I made a bracket in Fusion 360 to be able to fix the stick to the sides of the housing instead.
I finished by installing everything. I had to desolder the two analog sticks; they were too tall to fit in the case.
Everything works great! I can now kick ass by performing pro moves like this:
What remains to do now is create the back plate, and to make a front plate exposing the Plus, Minus, Home, Turbo, Capture buttons and the status lights.