Plan on private RV parks which is not going to be a major problem as you can drive into the parks and not miss anything. The US Forest Service and National Park campground will be very restrictive but often the BLM campgrounds are in more open areas where space is not at such a premium.
The challenge is in breaking up the trip to minimize the amount of driving you do on any one day. You should have a fairly straight shot up through Kansas and South Dakota and into Montana and the east side of Glacier NP. From there the decision becomes whether to come south through Idaho or head west into Washington. It depends upon what you want to see.
Check the temperatures in the high desert areas in the southwest as there is a steep decline going into November. I would hesitate to be planning to travel east late in the year when snow or ice storms could hit.
Areas like Glacier and Yellowstone are not at all crowded in early July and would make for warmer weather at the end of your trip heading back east.