Using a propane genny is still going off grid. Just fossil fueled not solar. I have a 2kw Honda converted to propane as well. Runs the aircon just fine.
The problem is noise. During the day we're mostly outside anyway so don't need the ac really. Its those hot humid nights....
If you're in pretty much any campground rather than pure boondocking you can't run your genny at night. National park and national forest campground quiet hours are 10p to 6a, and if you run your genny at 9p or 7a you don't have happy neighbors. Most state parks have similar rules if not more strict.
So, my objective is 12 hours of ac, 8p to 8a, with no input from a genny (or course there isn't much of anything from solar during those hours anyway). No way to do that without going to Li Ion batteries, lead acid is way too heavy.