There's a soft start kit you can install on your air conditioner that is claimed to allow a 2000 watt generator reliably run a 13.5 kBtu air conditioner.
Micro-Air - Products
My research into the product claims leads me to believe that this technology does indeed work. I've ordered one and will install later this spring so I can hopefully run the air conditioner reliably when it is super hot and/or we are at high elevation with our Yamaha 2400.
When dry camping, use your fridge on propane. I guess it doesn't hurt anything to let it switch to electric when you're plugged into the generator, but you're not saving any resources by doing it this way.
Yamaha and Honda have led this corner of the generator market for a long time. I don't have experience with other brands and they may work just fine. But you often get what you pay for.