Startup on these is spotty on a 15a circuit. My Roo runs on 15a but not all do as in your example. There are "soft start" capacitors that can be installed in the air conditioner that may allow the AC to run. But probably just that, nothing else.
I've known guys who just pulled the 15a breaker and put in a 20a but I doubt any licensed electrician will do that.
If you're going to the trouble to run a 20a circuit to power the camper you may as well run a 30a outlet. My current house has one with the same connection as at the campground and on it's own 30a breaker. Currently running the AC and anything else I want. Just confirmed the water heater works as does the microwave. My power panel is in the garage within 2feet from the sidewall so it was easy-peasy. If the breaker is in the basement it will require some wiring but worth the effort.
-- Chuck