I had a similar issue a few years ago.
I purchased a used TT. I built a rack on the back for my trusty Honda EU3000i generator.
When I plugged into shore power everything worked great. When I unplugged from shore power and plugged into the generator, everything worked except for the air conditioner.
I suspected that the generator had a problem so I took it to the Honda service shop. They tested it and found no problem.
Someone suggested that I install a "soft start kit". I had no idea what that was.
I found an RV parts warehouse nearby and asked them. They knew exactly what I was looking for. About $30 and half an hour on the roof installing it and the problem was resolved.
Good luck.
2004.5 Dodge 3500 5.9 Cummins, NV5400, 5" turbo back stainless exhaust, Edger programmer & 22.5 Alcoa's
2002 Forest River 36 5th Wheel (staying home)
1992 Jayco 29 5th Wheel (Mexico veteran & headed back)
2002 "faux" Wanderlodge 40' My new toy....