Ah, a fellow dry camper. The one time we did this trail, we ended up camping in the bottom of Dane Canyon because it seemed like everywhere the trail hit a ridge, there was someone car camping. If we did this trail again, I think we'd try for the flat spots between Dane Canyon crossing and Dane Spring where the trail is kind of on a ledge, above the water in the bottom of the canyon (it may not be 200', but you're not down in the canyon bottom) and a good ways below any roads on top of the ridge.
Unfortunately, most everywhere else away from water on the Loop has a road nearby (the road map of the area looks like spaghetti--way to do multiple use, Forest Service!). Throughout the loop, there are a number of dry ridge sections, but you will probably have car campers nearby--it might be less busy on weekdays if you can manage that.