I have clear polycarbonate bases (Lexan) 3/8" thick and they are rigid, rigid, rigid.
I receive them blank and drill them to suit. I get them mounted on the router first before attempting the center hole. Regular twist drills seem to work just fine but I always clamp a piece of wood on the backside when drilling. Never had it chip out, but never drilled it without the backer either.
Then I just put whatever's handy and pointy... like a countersink bit... into the router and lower the bit to contact the plate and make me an "unmistakeable centerpunch". Then I remove the plate from the router and drill the center hole in a drill press. And for that I've simply used a 1" hole saw. For smaller holes I use regular drill bits or anything I'd use for wood. I usally keep teh speed down because, as was stated earlier, the plastic WILL melt, but with the polycarbonate I think it would take a lot of heat. I haven't had problems doing it this way.
OH... and from building model radio controlled airplanes I have this neat tool that was used for centering the mounting bolts for engines. Cone-shaped, hollow business end kinda auto-centers in a hole. And a spring loaded drill slides through the cone to give you a nicely centered starter hole for mounting to your router.