A couple years ago we camped in the area, we were shooting for Chute canyon but due to the amount of rain and how late it was we ended up camping at the trailhead to Crack canyon - once canyon to the east. I feel like Chute canyon has a better camping area for sure, here.. 38°37'44.79"N, 110°45'47.52"W or just down the canyon a little farther in the wash.
You can count on a bunch of campers where around where Chute Canyon road and Temple Mountain road meet. My recommendation would be to camp farther west towards Crack canyon or up the North Temple wash, there are several spots that are amazing around this area that should be more secluded. 38.66927, -110.64810 The North Temple wash road (1012) can actually be followed all the way back to the Temple mountain road, it is seldom traveled and has a few rougher areas, there is some good camping nearby where they meet again, I have a recording of the track if you'd like. There are lots of pull outs with beautiful views along the Temple mountain road.
Tan seep has plenty of room for your trailer and should provide good level ground for setup, there are several places that would be suitable to camp when the Red canyon loop is in the wash on the northern end, also, Sulpher canyon has seen some camping for sure. The lower end (Hidden Splendor Rd) Is going to be a rougher area to find camping, so I'd shoot for one of the areas I listed above if you can, after the Red canyon road starts to head north, instead of West, you'll find a lot more camping options, it looks like some decent camping in the wash below the Lucky strike mine as well, although sulpher canyon has some old buildings that might warrant some exploration.. Anyway, hope this helps a bit, in short, you shouldn't have too much trouble finding camping options in the area, nor trouble finding seclusion. As far as camping goes farther north, I'd highly recommend "the wedge" aka The "little grand canyon" north west of the Buckhorn Draw.. the camping is amazing and only short drive from the petroglyphs. The views are worth the few people you might chance across, there is plenty of camping area so you can easily find a secluded area.