Today I received the new voltage regulator in the mail, so this evening after work I set about replacing the stock R/R.
First, I put the ohmmeter on the new unit. Every lead tested in the specified range, so I felt confident that at least the new R/R was in good shape.
Hooking up the wires was no problem at all. Since I got the prewired kit, I just had to cut my original stator wires and crimp/heat shrink them onto the new wiring. The kit came with leads that attached directly to the battery, so I hooked them up and then cranked up the Burgy. It purred like a happy kitten!

I tested the voltage on the battery: bike not running = 12 volts; bike running about 4-5000 rpm = 13.5 volts. That seemed reasonable, so I unhooked everything and started putting things in their places.
I bolted on the new regulator and ran the new wires behind the gas tank filler tube. I used a couple of zip ties to fasten them to the frame. Then came the fun part: putting all of the tupperware back on....
It took about an hour and a half to get everything back on the bike. I broke one tab on the little panel under the passenger backrest by forcing it a little too hard. That tab looks like just an alignment piece, so I am not worried about it.
While I had the thing torn down, I decided to put an emergency trunk release on the bike as well, so I drilled a little hole and hooked up a short wire with a keyring attached to the end. I tucked it in under the seat after I got the tupperware back on, and what do you know, it worked fine! It is kind of hard to fish the little keyring out from under the seat edge, but it can be done in an emergency.
Thanks again to all of you who gave advice and pointers about what might be wrong and how to go about fixing the problem. This is a great forum!
I'll update this thread again if any more excitement turns up from this episode.
Paul