I did that trip last summer
Arches, Bryce, Zion, South Rim. Great time.
We arrived at Bryce on a Tuesday, no problem getting a spot with no reservations, and we have a 32 foot trailer. Don't arrive on a Thursday or Friday.
At the south rim we stayed at 10-X campground, just south of Tuscayan, which is around 4 miles from the entrance. Again, arrived on a Tuesday, no problem, and some huge sites. At 10 bucks a night it was fantastic, elk in our backyard.
We did not drive the tunnel at Zion with our trailer. I think the limit is 50 feet (combined truck and trailer), and we are over that. We stayed just east the park. It was commercial, but convenient.
2008 F250 Diesel
Reese dual cam (finally dialed in just right)
Yamaha EF3000iseb generator
Renogy 100 Watt Solar Suitcase