I mounted my Happ comp joys in the following manner as suggested by Jack from ArcadesRFun, who cut out my panel:
"We install the joystick bases from underneath the panel by first drilling a shallow pilot hole smaller than the actual screw. We use 1" long zinc plated wood screws from Home depot or Lowes at about $4 per 100.
The size is #10. The box would then be marked 10 X 1.
The screw does not come through the top of the 3/4" panel as the joystick base is thick enough. We use only the outside corners of the joystick bases for the screws. You can test on a piece of scrap wood to feel confident about using that size."
I've had my control panel or a few years and it works great. With rigorous playing too.
Also, it's tough to line up the joystick holes to drill without this handy template from the real Bob Roberts: You can grab it here:
http://www.therealbobroberts.net/parts.html#joysticks (look under "Joystick Template")

It's only $2 and works great! An invaluable tool.
D