We've done this type of circle drive twice after picking up friends in LV. Here's my suggestion.
Day 3; leave GC at noon stay that night at Gouldings CG at Monument Valley.
Day 4&5; CG at Moab, explore the area, it can get hot in early June.
Day 6; CG in Capital Reef NP (Fruita) or Torrey
Day 7; Drive Utah Hwy 12 to Bryce (Ruby's). Make sure and stop at Calf Creek Recreation area and take hike to Lower Calf Creek Falls (this might be the highlight of your trip)
Day 8; Tour Bryce scenic views take a hike down into the canyon stay at Ruby's
Day 9& 10; Try to get into Watchman CG at Zion, explore the park via shuttle.
Day 11; return to Las Vegas