Since we're talking about extreme examples...
What the OP asked is possible. There are plenty of expedition vehicle owners that use a small-ish A/C unit (typically a household window-mounted type) in combination with large solar array, a bank of batteries, and a 2-3kW inverter to keep their rigs cool... and they never plug into shore power or run a generator.
I can and often do operate my roof-top A/C without starting the (gasoline) generator or connecting to shore-power... I just power it from my inverter. Of course, due to the modest size of my solar and battery systems, I am limited to about 3 hours of A/C run-time. After that I have no choice but to re-charge my batteries from a 115Vac source.
Unfortunately any the solar/battery/inverter solution is going to be far more expensive than the extra charge campground is going to have for power use.
-Mark