I'm suspicious that an air circulation/leakage problem might really be an issue even though things look alright. I took the ceiling assembly shroud off and moved the sensing bulb out of the return air stream and out in the open not under the shroud. The unit started taking an 8+ minute on-cycle and a 3+ minute off-cycle. (Still not as long of an off-cycle as I'd like) This is much better and with the new start capacitor, the starts are much easier.
I tied a thermocouple to the sensing bulb and read it with my multimeter. The thermostat appears to maintain a 3 to 4 degree deadband at 3 different temp settings. I think that is good enough.
So, I tried to run the AC on the generator. And what do you know, with an ambient of 94 and here at sea level, it handled it quite easily for 4 cycles no problem. Maybe today it'll get hot early and I can try an extended test.