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:
This reply is the closest thing to good advice in this thread.
The only thing that's going to get that out without cutting a slot is a tack weld. Which will likely burn the wood. Just cut a slot. You can find another aged carriage bolt to replace it.