I finally got a chance to tear down the rear brake today to try and figure out what was going on. I took apart what I guess serves as the master cylinder for the reap brake, the part that the pedal causes to pump the fluid. Everything looked pristine clean inside but I went ahead and hosed it out with brae cleaner. I didn't see anything that looked like a bleeder hole though. Put everything back together and once I got fluid pumped back through it seemed to lock up again. Well hmm. I decided to press down the caliper so I could at least drop in the new brake pads while trying to figure out what was up, and that's when I discovered the problem... the caliper itself was frozen.
Once I popped the caliper apart I saw a ring of sludge near the top of the piston. Tried to clean it up with a wire brush but then I noticed that there are actually gouges in the piston forming a ring about 1/4" from the top, and they look fairly old. Once I got it cleaned the piston slid back into the caliper smoothly and I was able to put everything back together again.
The brakes, with the new pads, are working perfectly at this point. My theory is that a previous owner ran the pads down to bare metal at one point, causing some chattering which created the damage ring in the piston. After I put the bike back together with new pads it was working fine until the pads wore down to the point where the damage caught against something inside the caliper, jamming the piston and preventing it from sliding. Now that I've put new pads back in again, the piston is pushed in further and not getting caught (at least until this set of pads wear down again). It will probably work fine for awhile, so next time we have a nice day I'll go for a ride and burn through the full tank of gas I have, then I can park it for the Winter.
Since I don't know what kind of damage might also be inside the caliper, I think the only plan here is to just get a whole new caliper assembly and scrap this one. At least it is easy to swap out.