Don’t be the person at the campground with the cheap, unmaintained, builder-grade generator. Just. Don’t.
“But it’s the best/biggest thing I could afford!”
You’ve approached it by trying to solve the wrong problem. The problem is not, “How much wattage can I get for $1,000?” It should instead be, “How much generator do I actually need balanced with how badly do I want to annoy the hell out of everyone around me?”
Save a few more dollars and buy the best-quality, inverter-style, eco-throttle, RV generator available to you. You and your neighbors will be extremely grateful.
And, yes, a 2KW generator really will run your RV’s air conditioner.
After having to listen to the neighbors on both sides run their 9kw Champion and 8kw Honeywell generators for the entirety of the non-quiet hours and well-into quiet hours, I find myself pondering the plausibility of a remote generator-disabling mechanism.
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For my own, future reference, I’ll leave these here:
My #1 pick, if I can find a good spot to mount or install it is the Honda EU3000iS:

I wonder if it could be fit into a 5th Wheel’s typical generator bay.
I hear (hah!) it’s the quietest generator in its class. There’s one here at the campground that I walked by and I’m quite impressed with it.
There’s also a nearby site with a Yamaha ES3000iSEB, which, too, is quite nice:

There are, of course, built-in options available from Onan – maybe the RV QG 5500 LP would be well-suited to the task.
I think the Honda is the way to go. Prices vary. In my experience, paying a bit more for better quality up front, and caring for machines and implements properly will greatly reduce their overall cost of ownership and improve their reliability and usability in the long run.
For now, I’m going to try to enjoy the 100F heat without resorting to running my own generator and contributing to the noise pollution.