Be optimistic about your future: assume that you will get another shot! ;)
That many miles would force you to do nothing but drive for 9 solid days out of, what, 14? You would be driving past some of the best memories they could have, of time spent relaxing and enjoying quiet times around the campfire, hiking the trails, seeing the critters that pop out of the grass 10 feet away, seeing the stars, etc. Plus they might go stir-crazy in the vehicle, and start driving YOU crazy.
Don't get me wrong... I have traveled like that myself, alone. But I had no one to please but me. And I've learned since then that moving fast and touching just the edges of the highlights is not much (if any) better than slowing down, lingering, and savoring. The fast jaunts gave me enough overview of places that I now know what I like and what I might expect as I pick an area and explore it in greater depth.
I agree with Hank, Custer SP and Rushmore area is great. The national forest campgrounds north of Rushmore are pretty nice, in the evergreens. Drive through Deadwood, and see if you can resist stopping to explore this (admittedly touristy) western-style town.
Look up info on Thermopolis WY. Interesting hot springs to soak in, and see the colorful mineral deposits. Take time to drive the Beartooth Hwy just north of Yellowstone NP. Linger a while in the Tetons, where you can ride the Snake River (smooth sections or white water), hike around Jenny Lake, and enjoy the view of mountains that seem to rise up close by from the flatlands.
Still have time and a desire to move? Head down to Green River and see Flaming Gorge. Farther south is Dinosaur Natl Monument. Then near Grand Junction, don't miss the drive through Colorado Natl Monument. But on the other hand, if you find plenty of enjoyment with S. Dakota and Wyoming, on the way back you can stop near Laramie at Vedauwoo and climb around the rocks for a couple of hours.
Hey look, the Grand Canyon is just a big hole in the ground, right? ;) Just keep telling yourself that, until it's time for the next trip. :D
