When I had to take two 1" cuts off the screen of my GL1000, and the same operation on my Silverwing, I used a variable-speed jigsaw (plunge cut saw) operating at the VERY lowest speed, with the cut marked and taped with paper-based Masking tape.
Try the same with drilling - variable speed drill at its LOWEST speed (drill bit with very little undercut), and mask the hole on both sides with paper-based masking tape. Using the absolutely lowest speed means that you can actually stop drilling if the drill "picks-up" in the plastic.
The real problem with drilling plastics is the heat produced - it melts the plastic, and this stops the drill bit.
Slow and steady wins out. If you think the platsic is about to melt - STOP - and let things cool down.
It's also a good idea to hold the drill fairly loosely in the chuck, to allow it to slip if it bights hard.