1-1/8" center hole should be fine, but go 1-1/4" if you want to be extra safe. I would mount those underneath with no routing. That's how I've got my P360s and Competitions mounted. Some of it is personal preference.
The best suggestion I could make is for you to contstruct a small stand-along test panel. Mount one joystick with no routing, and then mount the other one with 3/16 to 1/4" of material routed out (assuming you are using 3/4" material). Play both of them and see which one is more comfortable.