Given the time frame and season, I try to see Glacier National Park, then Mount Rushmore and then drive east through the Badlands National Park. That will keep you north as you head east which should give you the cooler weather until you start to head south around Minnesota.
Trying to go south first on the west coast before turning east will put you into some really hot weather next month.
If you still have the time once you get to the lower 48, you might try to drive through Yellowstone and the Grand Tetons.
You have a lot of possible routes that you could adjust on the fly given how far your total trip is going to be. Good luck on your PCS.