I'd consider it if I had a stock set up too. Keep in mind, of course, that the dry weight doesn't include the weight of the battery or propane, along with any gear you put inside the trailer, or any gear/passengers in the Jeep itself.
The class III hitch is rated at 500/5000 so you are good there.
I did use a prodigy brake controller towing a 17' 3500lb camper. The controller worked great, it applied the trailer brakes a split second before the LJ's brakes, thus the camper did most of the stopping. I did have one issue with the controller, however, in that it had an adjustment wheel that would occasionally vibrate to a higher setting and that caused my right front caliper to press against the pads when I was cruising at highway speeds.
I also recommend a weight distribution hitch (I use a Reese). It spreads the weight evenly across all three axles, and that made a BIG difference.
01 XJ Sport
06 LJ Unlimited Rubicon
08 XK Commander