After carefully and gently clamping the top down, I started by drilling a few pilot holes, and then I used a small keyhole saw to cut them. No high-tech stuff here, just old-fashioned hand/eye coordination. I used a template to make sure the holes were symmetrically placed, that's about as technical as I got. I used an x-acto blade to refine the cut, and sanded them smooth. They just need a little more cleaning up and they'll be good to go. Since I'm going for the look of an old radio design I have decided not to put binding on the inside edges of the holes.
Voila, I now have enough room at the back end to place the buttons - no more pesky stuff in the way of strumming problem.