We have a 2kw generator, but have yet to take it on a trip. Instead we make do with a combination of campsite selection (shade where possible), fans (ceiling and a little portable USB box fan), and altitude (as pointed out, higher = cooler). Since we’re in southern Arizona, it take a day’s drive to reach elevation. If going further, such as US 395 in California, we bite the bullet and stop at RV parks with electricity for a night on the way to cooler climes or if forced to it, a motel.
Main reason we’ve not taken the generator is where do we put it and a couple gallons of gasoline? We like our minimal footprint – don’t want to hang a large platform off the back hitch – we’re not wild about hanging it off the front either, and are leery of putting the generator/gasoline in the back seat which is already pretty crowded. I know, picky, picky, picky!
I have an old ford truck (1982) It has two fuel tanks (front 19 gallon)(rear 38 gallon) I just use a squeeze pump to fill the generator. I guess you could consider mounting 5 gallon jerry cans on the rear of your camper to bring extra.