Going thru the dessert anytime after the middle of June can be brutally hot. Even in late May, my brother and I would hike, or travel in the car, from 5 to 9 in the morning and 5 to 9 in the evening. It's good that you taking the northern route on your later leg of the trip, when it will be even hotter in the dessert.
Bare in mind the you can be miserably hot, and freezing cold 15 minutes later when the sun goes down.
Back to the Burr trail. There is a campground near the Burr Trail. ( next to a petrified forrest, maybe near Escalante ) You could unhook your trailer and check out the Burr trail with no problem. I enjoyed it.