Update:
I've added
bold comments to my original post at the top with the answers I have come up with so far. The motorcycle has been running great with just a super capacitor to start and run the bike. However, after about 3 days of letting the bike sit, the charge drops enough that it will not start. Anything below 10v will not start the bike.
Here's a video showing how low the voltage can go on the bike before the computer stops functioning and the memory resets.
I was concerned that no power is generated from the alternator on the bike below 2000 RPM, but was surprised to find that it maintains 12v at 1000 RPM idle with a single 55w headlight on. Here is a video of that experiment.
Here's a general video with me explaining my setup, and showing the bike starting, the peak amp draw at start (about 80 amps), and subsequent charging.
As an aside, I fixed the ATV starter by installing a new brush plate, brushes, and o-rings. The other brushes were badly worn, oil was infiltrating the motor, and I had accidentally shorted the positive post of the motor to ground. No testing has been done with replacing the battery with a super capacitor since my dad seems to be happy with the battery from my motorcycle. When I get more time I'll swap the capacitor in and test it.
EDIT: After 1 week of sitting, the bike voltage dropped to 7.5v. I coasted down my driveway and easily bump started. The computer retained all memory.