In fact, I went and did it right after I posted. Took about 45 minutes, and at least 20 of that was spent looking for my cordless Dremel tool. I gave up, and brought down the plug-in Dremel tool, which had an flexible shaft attachment with a flap whell in it, so I started there.
Buffed with the flap wheel:
The legendary "TIX Solder" and flux:
And for a source of Thermal Energy, I have my trusty Weller SP-80, 80-Watt Soldering IRON:
This ain't yer Daddy's wimpy Temperature Controlled Soldering Station! No siree, this here's a gen-you-eine Soldering IRON, and it has a SOLID COPPER tip about the size of a 50 cal slug:
It's 2" long, and a half-inch in diameter. In the battlespace of "Heat The Connection As Rapidly As Possible", it wins handily in this application.
Applied the flux after cleaning the brass again with an alcohol wipe, plugged in the iron, and went for a cold soda from the fridge. Applied the iron to heat the metal, and then flowed in the solder. After cooling, I cleaned all the flux reside off, and carefully re-tinned the tip of the iron with plenty of rosin-cored solder to get any of the other flux off. It has chloride compounds in it, and they're very corrosive if not cleaned off.
Ready to return to service:
Hey, I broke it, so I get to fix it!