The 1-1/8" hole should be more than big enough.
How to install them will depend alot on personal preference.
You CAN just screw them to the bottom of the CP.
If you have the stock bat-style handles, this should give you a high enough stick.
If you have the short balltops, you will probably need to rout them into the bottom of the CP a bit.
The long balltops come out a bit short for most people if you just screw them to the bottom, but I actually like them that way myself.
I would suggest that you drill a hole for the stick in a piece of scrap wood, and then screw the joystick to the bottom of it.
Play around with that for a bit, and decide if it feels OK to you.
If you find the joystick too low, you can rout the base into the piece of scrap wood a bit, and try it again.
(I would start with routing 1/4", and then go 3/8", and 1/2" absolute max., if necessary).
The 1/2" won't give you much left.
If it STILL feels too low, you might try another piece of scrap wood, but rout the joystick into the TOP of the wood, so that it flushes up with the top of the piece of wood.
If THAT feels too low, you're pretty much out of luck, as that is as high as the stick will mount nicely.
That said, I solder ALL my connections (except the screw terminal ones, obviously).
Soldering is a personal preference; but I've NEVER had a solder connection go bad on me, and have saved ALOT of money buying my $3.19 roll of solder vs. all the quick disconnects I would have had to buy over the years.