We've had the Yamama 3000 (NON boost model) for 8 years, and it runs our 5th wheel toy hauler 13.5k A/C with no issues. It can run everything else at the same time (fridge, chargers, lights, etc) with no problems at all. If it was a problem of not enough juice on the generator - usually the protection kicks in on the generator before anything happens inside. If your A/C fan is still running but the compressor shut off, you are probably right the thermostat was set odd or you have some other problem, but doubt the generator is the issue (the fan runs on 120v, so if the generator pops the fan will stop also).
Well test confirmed, AC runs fine once I turned the thermostat down Generator was humming out in the Alabama heat like a YZR-M1 at a fast idle but of course under 60 dBA