Remember also, that with the Phantom, the controller is a Teensy 2.0 soldered to the underside. So, a layered case would either be very thick or have a step. You would also have to figure out how to route the cable and if you want the cable to be detachable.
I attached the original plate CAD files, because I don't remember where they were I downloaded them...
The plate should be 1.5 mm thick, +- some tolerance. The vertical dimensions should be in the Cherry MX data sheet.
You could route a step in the case to make a ledge for the plate to sit on or let the plate be a full layer in the layered design - either way, you would have to make the original plate design wider. The ordinary plate design is only a mm or two wider on each side than the PCB and I think that would be too narrow for wood.
You could also put cylindrical spacers in the holes of the PCB holding the plate up.