Most of the nuts came out with minimal force, but the last one just spins in place. I tried rubber gloves, painter's tape, and duct tape to get a better grip. I'd rather not cut a slot in the domed nut if there's another good strategy.
The rest of the restoration underway:
The hole does go through to the other side, right? So you probably already tried applying force from there? Perhaps you could try cooling it down (so it shrinks a tiny bit and hopefully gets loose)