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:
Try clamping a piece of wood to the handle on the domed nut side with a bit of neoprene rubber or a ball of duct tape wedged in between. If you need a stronger bond that is still removeable, try something like hot glue, silicone caulk, E6000 glue, or super glue.
The clamping approach worked great!