So I got an extra board to experiment with, and I found with zero voltage in the Supercap, and I add in the battery, the voltage stays steady at 0.90 volts, and the ISS is transmitting. Remove the battery and of course it drops over time, I did not watch where it stops transmitting. Now with the TI chipset being a 2.0V chipset, that would have to say there is a boost converter in the circuitry somewhere. I see a couple coils in the circuitry, whether that's there for buck/boost I dunno.
But still interesting. I'll look at it further.