The crappy RF switch finally quit working for good, so I was forced to drill the hole to mount the air pressure button.
In a solution that Solomon might have approved of, I did none of the suggestions offered above. My wife wanted the switch mounted away from any of the pre-punched holes, so I drilled a hole in the quartz with the diamond hole saw (incredibly easy), and then went out and bought a carbide-tipped hole saw and drilled through the stainless-steel sink with that, using the hole in the countertop as a guide. I used a lot of WD-40 as lubricant, and the new hole saw drilled through the metal pretty easily. The end result looks like it was installed professionally, and the air switch works great.
The diamond-grit holes saw kit that I bought came with a foam storage device, that also doubled as a guide and water bath for the bit. It was a very clever solution to cooling the bit.