One more comment. I'm still concerned we're going down the wrong road thinking that the gen is shutting down due to an overload. If I was there, I'm pretty sure I could tell if the gen was shutting down due to a short overload. The gen would bog down quickly, and it's likely there'd be a spark as I threw the gen breaker on. Can you tell if that's what's happening - an overload type issue? It's hard to come up with other type problems that match your reported symptoms, but not impossible. The gen probably has remote start/stop. You might even have autostart/stop on the gen. There could be problems with those that show up only when power is on.
Still, I'd be ruling out the ATS first, since you've done most of the work needed.