Let me preface this response by saying that some people on this board and other towing forums are of the mind-set where if you can hook your tow vehicle to it, you can pull it. I, on the other hand, defer to the engineers that designed, tested, and generated the ratings for the vehicles. I trust that they didn't pull the numbers out of thin air, and actually spent some time doing R&D on the components affected by towing. (I've witnessed it myself first-hand)
Now, that being said....
Your truck is rated to tow a trailer weighing 3240 lbs for a 2wd or 3380 lbs for a 4wd. Regardless of the class-III hitch installation, that alone doesn't change the rating for the vehicle. Furthermore, that rating is based on a 7700lb GCWR for the 2wd and an 8000 lb GCWR for the 4x4, and assumes a 150 lb driver and nothing else. For every pound over that you put in the tow vehicle, you lose a pound of towing capacity. For every pound over 4460 (4620 for a 4x4) your Explorer weighs, you lose a pound as well.
If the published dry weight of your trailer is 4000 lbs, I'll bet that even with dry tanks, you're still looking at nearly 4500 lbs after loading your gear alone. Typically the camper manufacturers don't include the 'optional' items in their dry weight, like the battery (50-60 lbs), propane tanks (36 lbs each), refrigerator, furnace, AC, awnings, water heater, shower, etc. Add the 'optional' equipment to the gear you add into it, and it's amazing how quickly your 4000 lb camper becomes 4500-5000 lbs. (By comparison, our Coleman popup has a 'dry' weight of 2400, but it tips the scales at before we loaded our gear, and 3220 when we hit the road!! {empty tanks, of course})
IMHO, your Explorer is just not the right tool for the job. You're going to be putting a lot of strain on the transmission and rear end. The weight distributing hitch will help prevent overloading the rear suspension, but the drivetrain is still going to be over-capacity by a fairly large margin, even if you're at 4000 lbs fully-loaded. Having towed a 25' boat weighing 4650 lbs with my Explorer (which was, in fact, just barely within the specs for my rig) I can tell you that it's not going to be a pleasant towing experience. It's gonna push you all over the place.
Now, that being said, a HD trans cooler will go a long way towards helping the trans, but there's no way to cool the rear end and that gearing is gonna have the trans working hard just to maintain speed. Flat-land, I *might* try it a couple of times, but as a long-term solution, it's just not the right tool for the job, IMHO.
How's that for an ambiguous response?
-Joe