Here is the latest report on this issue. I decided to address this problem of the Super Cap apparently not doing anything by replacing the ISS transmitter board. I could have replaced the Super Cap but I just didn't feel like figuring out how to solder standing at the top of a ladder. Like others on the forum, I bought a replacement board from Ryan at Scaled Instruments for about half what Davis charges. Swapping that out was supposed to be fairly straight forward. And it was, except for the discovery that when I moved the anemometer last summer, I used weather-rated Cat 6 cable. I spliced that cable back into the Davis telephone wire inside the ISS plastic box and it was the dickens to pay to manoeuvre those 4 weather-proof splices back through the cutout in the box. But eventually we (my son and I) were able to coax the splices out, one at a time.
After we got the new module hooked back up (it was much easier to thread the splices back into the new box than it was to take them out of the old one), we took the console out of setup mode and discovered that although we were transmitting, there was no external data on the console. We double checked everything, including making sure all the cables were tight, in the right ports and installed a new battery. Still no data.
I called Davis. They told me everyone in the world (except Edouard) used the factory default of Station ID 1. They had me change the DIP switches to use Station ID 5, then change the console to ID 5 and this time, we had data! They told me that changing the Station ID was a frequently-used trick to get the ISS to restart transmitting properly after swapping out the transmitter board.
So my big adventure turned out fine. Now I will wait 5 months and see if my battery dies again. If it does, then I will have to swap out the solar panel.
Whilst I had the thing apart, I cleaned out the collector cup, applied Rain-X to it and blew the spider webs out of the tipping buckets, etc. We are supposed to get an inch of snow tomorrow so I figured that my annual maintenance should not be procrastinated any longer.