I've been on the White Rim as a pickup truck supported mountain bike trip. Absolutely incredible scenery. We used a stock Chevy Silverado without difficulty. Much of the road is easy dirt and sand. Here and there you encounter a rock step that is a little steeper. You also have at least one spot that overhangs that could cause problems for taller truck campers. But the biggest worry with the trailer is that there are a couple spots with steep, tight switchbacks. Not sure how that would go with a trailer. Hopefully someone that has driven a larger portion of it will offer some advice. I rode most of it, and it was awesome and not at all hard as a mountain bike trip (unless you do it in one day).
I also recommend lots of beer.