Looks like a great trailer for an awesome truck. Just a few thoughts...
Good choice on single axel trailer. That is going to save significant weight and you don't need the extra carry capacity with your 5k vehicle limit. If you were going to go mini toy hauler in the back of the trailer, say ATV or Motercycle, double axel for sure. But then you would be at the limit for your tow vehicle too. As a normal, lighter weight camper, no problem with single axel.
I would want the v8 4runner if doing a mini toy hauler.
Regarding towing with the v6 engine, keep your camper construction light weight to keep the overall load down. I would shoot for 3k maximum load. Your 4runner could take the full 5K load, but performance is going to suffer. As you have, trailer brakes and transmission cooler are a must. At 3k weight in calm winds, you will likely only drop one gear on most grades. No speed drop, if you live with the higher RPM on the grade. Say 2.5K RPM 5th to 3.5K RPM 4th to maintain 65-70 mph on grade.
(Hey, don't look at me like that! Nevada has open road 75 MPH speed limits, with no reduction for trailer hauling. Oh, how I hate traveling in California with the mandatory 55 MPH with trailer!)
Then the only thing you have to worry about is wind load. That will be a challenge, but shouldn't be horrible. In a headwind or cross wind, you are going to feel it! That is just the nature of a box trailer. I would expect to drop two gears, from 5th to 3rd, when going up grades in a head wind. That is going to drop your speed to 55 MPH at about 3.5k or 4k RPM. There is no way around it. Just be glad it is not a bigger box!
I had a v8 4runner. I loved it! I was sad when I had to let it go during the economy crush, but I am also thrilled with my FJ now. V6 FJ and V6 4runner have similar engine, and same towing capacity. I don't even know my 1750# teardrop is behind my FJ, except on grade. On grade, it drops one gear (5th to 4th) and holds that pretty easy. Also only dropping one gear, it holds speed, albeit at higher RPM. So far it has never dropped to 3rd... But I expect your 4runner, with heavier and more wind resistant trailer, will.
Bottom line, v6 4runner should do great! At about 3k load, I wouldn't even worry about it. As you approach 5k load, I would probably install premium high temp resistant transmission fluid and keep a close eye on transmission temperatures just to insure long life.
Hmmm... I did a little more research. 2002 4runner 3.4L v6 is 183 HP 217 ft-lb torque at 3500 RPM. Also a 4 speed automatic. (No 5th gear overdrive) Not exactly a beast. For comparison, FJ is 260 HP 271 ft-lb torque at 4.5k RPM.
I revise my estimate. At 3000 lb trailer, you are going to commonly drop one gear down to 3rd gear on grades and drop to 55 mph. You should still be able to hold that in headwind.
But that means at 5k trailer you would be 2nd - 3rd gear cycling at 30-45 MPH on grades... Shudder!

Just keep the overall trailer weight down around 3k.