I have a Champion 3500/4000 and it will run my 15K AC. I turned off everything but the converter, but the battery was fully charged. Ran it for about 2 hours fine just to test it before hurricane season. It was about 90 degrees at the time. I just turn off the fuel supply to run all gas out of the carb on my small engines that will be sitting a while. Never a problem.
You should be able to run the AC. It could be that the starting capacitor in your AC is bad. This could stop the generator but the shore problem would have no problem overcoming it. It could also be a problem with the generator. Test the generator by running hair dryers and other high draw appliances.
