On the plus side, your troubleshooting so far seems logical and provides good information to work with. As for your dealer???, I know a lot of folks running 13.5k air conditioning units on that particular generator.
Don't know what the comment on the starting capacitor was all about. My timber ridge tt had a coleman 13.5k ac unit that wouldn't start with two paralleled 2000 watt generators until I added a start capacitor. Why soft start caps aren't standard equipment on all units is beyond me. They can extend the life of the compressor motors as well as making them less demanding on your generator.
As for an easy way to test the generator to see if it is having issues, what other means do you have to put a load on it? Being a 3000 watt unit you'll want to use the 30 amp plug, maybe power up your trailer and try running maybe a hair dryer, and some other high drain item on a separate circuit breaker inside the trailer. The problem will be having a good idea of how much of a load you're actually putting on the generator. If you have a kill-a-watt it would give you a good idea of what individual things you're plugging in draw power wise. Access to a load bank would be really good, but unfortunately I only know a few folks with access to those.