Would you believe this is what I tow.......
Even towing the lightest Coleman pop-up, I had to 2nd gear up some of the mountains around here.
But......as you can see I have the older TJ, and it has the 4 banger. As others have indicated, I think JKs with the 6 cylinder with the 4.10s is rated to tow 3500 lbs.
The hitch may be a concern. When I checked out the JKs a couple of years ago, the hitch was a class II. Class IIs usually just have 1 1/4" openings, but Jeep decided to put in the 2". That can be misleading to folks not familiar with hitch ratings......."if my 2" shank goes into it, then it should work".
There should be a sticker on the factory hitch with the hitch ratings. Most Class II hitches have a 3500 lb. maximum tow rating with a 350 lb. maximum tongue weight. Some A-Frame tongue weights may exceed 350 lbs. when the camper is setup ready to travel.
Most (if not all) Class II hitches do not allow for a WDH setup.
You guys are great with talking this thing out, and realizing the limitations of pulling with a properly equipped Wrangler. Y'all are pulling pop-ups and A-frames, not conventional full sized campers.