I'm not sure how a mounting plate would help? where does the plate go and how does it help in this situation?
The mounting plate creates a mini metal control panel in the middle of the wood panel. You cut out a square big enough for the joystick to pass through, mount the joystick to the plate, then mount the plate to the panel, hanging the joystick through the hole. For a flush surface, you can route out the thickness of the plate in the wood panel to recess it.
Your problem, however, is that the panel is already laminated. If you don't want to re-laminate, you'll want to go with an attractive plate. If you're using black laminate, a black mounting plate will look fine against it and you won't need to redo everything.