Litster's case design with integrated "plates" was not made for the switches to clip to the plate. They are just force-fitted in there.
You won't need to CNC-route any thick plates though. You could instead cut all of that out. I see two options:
1) If you instead use switches with fixing pins then those would mount sturdily to the PCB without the need of any plate.
I see that those white PCBs even support Cherry's PCB-mounted stabilisers if you would want them.
2) Get metal plates from
Falbatech. They were not designed to be part of the case sandwich but to be inside the keyboard case in-between its walls. The plates seems to be the right thickness for switches to clip into them.
Falbatech also has a 80-key version (Thumb keys are all 1×1 - sized).