I tow with an '06 V8 Mounty (4th gen) with HD tow package. Drivetrain specs are similar to your '05 V8 Explorer (3rd gen), except my 6-speed auto has a much lower 1st gear, which allows it to get the full tow ratings with the 3.55:1 gearing. But yes, you're numbers look correct. Your 5-speed with 3.73:1 gears and V8 will tow a lot more than that popup. I tow a 5200 lbs Surveyor hybrid just fine.
The 5000/500 rating you see is what you are limited to without using a WDH. The biggest component for that is the 500 TW limit. More than 500 lbs will remove too much weight from the front wheels. Adding a WDH kit puts that weight back on the front, giving the full 7000/700 rating.
But for your popup, dry TW is 314 lbs.
Typical loaded weight for popup is 600 lbs over dry weight. Let's just say 3000 lbs for you. At 12.5% TW, that's only 375 lbs.
If you maxed things out with 3314 GVWR and 15% TW, that's 497 lbs loaded TW, right at that limit. But I guarantee you will never get 1000 lbs of options and cargo in a standard height popup.
I recommend you start with a regular drawbar of sufficient drop for the popup. Then, as someone else mentioned, add a simple friction sway control bar. Any hitch shop can easily set you up, or if you are handy you can do it yourself. Harbor freight sells them for about $50.
After a few test-tows, if you feel that the TW is taking too much weight off the front wheels (headlights in the sky, or steering feels squirrely), you can get a WDH kit.
-Reese mini-350: Not enough TW rating for you.
-Reese/Drawtite 400 Single Bar: This is what I used for my 2000 Coleman Mesa, behind a Chrysler minivan. I loved it, and was able to continue using the friction sway controller since the head was predrilled. As long as your TW stays under 400 lbs, this is my recommendation.
-If you want integrated sway control, and/or TW could be over 400 lbs, consider the 600 lbs Equalizer 4pt WDH kit. This line is probably the most popular on this and other camper sites. As a side benefit, it would give a bit of growing room, in case you want to upgrade to a high-wall popup, or a very lightweight travel trailer. Remember, you have a lot of towing capacity to spare.
thebrakeman ('70), DW ('71), DD ('99), DD ('01), DD ('05)
2004 Surveyor SV261T (UltraLite Bunkhouse Hybrid)
2006 Mercury Mountaineer V8 AWD Premier
Equal-i-zer WDH (10k), Prodigy Brake Controller