We just completed a loop west from Illinois - 5,300 miles, 3.5 weeks including SD - Custer State Park, Badlands Nat'l Park, Mt Rushmore; ND - Theodore Roosevelt Nat'l Park; MT - Glacier Nat'l Park, various nat'l forests & Yellowstone; WY Grand Teton Nat'l Park; then visited family in Denver and back to Illinois. This was mid-Sept-ish and campgrounds were FULL - we couldn't get into several campgrounds. Rangers unanimously said they couldn't believe the crowds after Labor Day. Not sure what that means for next year.
You'll be driving right by Jackson Hole/Grand Tetons Nat'l Park - between Bryce & Yellowstone. Might want to stop for a night or two.
Be sure to check opening dates for places you want to visit - some of the campgrounds on the North Rim of the Grand Canyon and Bryce weren't open in early May when we were there, can't remember the year.
Personally, I'd try to make reservations anywhere you really want to stay - even though we hate being subjected to reservations. In the Mt Rushmore area, do not miss Custer State Park (again, make reservations) - in our opinion, better campgrounds than the commercial ones closer to Mt Rushmore & the drive up Iron Mountain Road is beyond scenic - use the toad though, the motorhome might not fit through the tunnels.
And a final comment - That's a huge expanse of country to cover in 3-4 weeks, might want to add at least a week.