"At worst, attempting to drill through the steel rivets will mess up a drill bit, but I have enough of those."

Use a drill press, not a hand drill. Clamp the work securely and back it up with a hardwood block or similar to avoid the bit chattering. Use a smaller bit to drill a pilot hole centered on the rivet. Use plenty of lubrication, a sharp bit and go slow. Use of titanium-nitride coated bits or cobalt bits will help to minimize bit dulling.