The following pertains to the first version of the circuit.

The heating system was in use until June 1st, so I did not want to take the chance of screwing something up and then trying to fix it in panic mode. I decided that I did not really need another super capacitor laying around so I rewired the two 350F super caps used by the previous UPS from series to parallel and connected them to the new UPS. This also made the mechanical changes easier (i.e. no mechanical changes). I had to discharge both super caps completely before messing with them. Here's the new UPS installed into the enclosure for the heating system controller:

The previous UPS was wired directly into my data acquisition "Hat". This UPS connects via the normal microUSB for power. Note the bright pink cable that I got from the dollar store...for a dollar!

I had to change the GPIO assignments for the powerfail.py code since GPIO24 was already in use. I let the supercapacitors charge to full voltage overnight and then pulled the plug on the wall wart. It went through the shutdown procedure like clockwork and shutoff the power to the Pi. When I plugged the wall wart back in power was reapplied after 5 seconds. Mission accomplished.