Wanted to run my AC with a Honda 2000 but it would not start the 20 year old 11k dometic. Put a Suppco in there they said, one guy even guaranteed it would work.
Son of a gun it worked! In cool weather, someplace between mid 70's and upper 90's it quit working and would fail to start.
Long story, read a lot of misinformation on the net and changed to an 8k window ac, with a hard start. The mighty little Honda runs it under all conditions. Sea level, Mountains, Death Valley, high humidity, never fails to start.
So in my WAG opinion, either a smaller AC or a larger genset is needed. After reading and if I wanted a bigger AC I would try the Micro Air soft start.
I like what I have read about the Champion generators, if I was going to buy one I would get the electric start as going out to start and stop the genset gets old.