The 3500/4000 are generally considered as dependable as any other generator out there. That's not to say that this unit can't have a issue and a exchange or call to Champion may be in order. I wish I knew how to test it to see if it was putting out it's rated output. With a 4000w peak there should be no issues running a 13.5k AC at sea level.
Now if it's the AC somehow then a weaker inverter generator with 3000w peak isn't going to help no matter how quiet.
Without sometthing like a kilo watt to measure the starting load of the AV, the only way to simply test the generator would be to plug in the RV to the 30a outlet, put a 1500 watt heater in the trailer and plug another into the standard outlet on the generator. running both should be 3000w and that's higher than I was told it would take to start a 13.5 AC. If it will run both it's time to go back to the AC.
Possible the capacitor isn't doing it's job and providing the needed assist for starting the AC meaning it takes more than usual to start it. A larger generator would handle the load and even a 20a circuit can take a pretty good punch for a short time before popping. They may be hiding a problem that has just started or may have always been there.