No problem I thought, I'll just replace the nuts (and use some Loctite). But once I looked up underneath the bike, I realized that wouldn't be happening without a whole lot of disassembly. Because of the subframe, it's physically impossible to reach the attachment points. Hell, I couldn't even see one of them with the fuel tank in the way.
So, after a little research, I replaced the whole thing with this:

The cool thing is this - the factory tail light assembly has the bolts as part of it, with the nuts holding on from inside the fender (the area I couldn't reach). BUT! this Vespa tail light is different; it's on the receiving end of the bolts. All I needed was a pair of bolts protruding out of the existing holes in the fender.
So I snaked some string down through the holes, down to below the bike, tied some bolts to it, and carefully pulled them back up through the holes. I stuck some teflon washers on the bolts to keep them from falling back out. Then it was just a question of bolting up the new tail light bracket, wiring it up, and screwing on the lens.
Bingo! Not only is my problem solved, I also now have a dead-sexy new tail light!