Being faced with the similar things here I more or less changed my priority on getting a generator to run my air conditioner when camping off the power grid. Here on the East side of the US seems like all of the camp grounds have generator run time restrictions anyway and we don't have many private or dispersed place to go to.
My 2KW generator requirement is purely for re-charging my battery banks and we get to run all of the 120VAC items we want to have on using an Inverter and 12VDC items direct connected to the battery bank. At 8Am the next morning we connect our trailer directly to our 2KW generator 120VAC receptacle using a RV30A-15A adapter and re-charge our battery bank back up to its 90% charge state so we can do all this all over the next day/night camping off the power grid. Does not include air conditioning or high wattage microwave but just about everything else we want to use runs from the 12VDC battery bank.
My brother-in-law from central Illinois has the two 2KW Honda Generators for his fifth wheel setup and about the only time he gets to run his air conditioners is sitting on the side of the road somewhere cooling things down to get a quick snack etc... He uses the electric hook-up most of the time at the camp grounds to run his air conditioner and high wattage microwave...
We camp alot off the power grid and very seldom find a place where we are allowed to run our generator here on the East side of the US except for a couple of hours in the morning and maybe three hours in the early evening. We have been at some Natl Forest areas that allowed the generator to used from 8AM to 8PM each day but never after 8PM anywhere we have been.
The NASCAR places and private owned non-public camping spots I guess you can run one 24/7 ??
Or if own one of those $300K motorhomes that has built-in generators seems like they get to be run anywhere they are at. I hear them running at campgrounds all the time that won't let portable generators be run except during allowed generator run time periods. Money talks I guess..
We get by just fine running everything off our battery banks and recharging every morning with our 2KW generator when allowed to run the generators. Having solar panels would really help out too keeping the batteries re-charged during the HIGH SUN days...
Roy Ken