Heating and Cooling on the road

I would never say "stuck with the generator". That's what it's there for. Your own on-board power supply to run AC, TV's, etc. One of the reasons I bought a class A was to have a rolling apartment and all the comforts it offers. So yes, generator. We run ours pretty much 100% of the time when travelling. Especially in the summer. My wife and son cook food in the microwave, watch TV, all while the central AC keeps us cool. Fuel impact is negligible. Might detract .5 mpg. Your dash AC will likely hamper your engine that much and won't offer any cooling to anything 3 feet behind you. Your inverter will not run the AC.