DW and I did a similar trip with a car and tent gear in 1998. We only had 23 days on the road.
Badlands, Mt.. Rushmore, Crazy Horse, Devils Tower, Yellowstone, Grand Tetons, Jackson Hole, Lake Tahoe, San Fran, Yosemite, Sequoia, Kings Canyon, San Diego, Las Vegas, Zion, Grand Canyon, Bryce Canyon, Arches and a quick stop at Rocky MT NP on the way home.
We missed Pacific NW but were able to see some redwoods near SF at Muir Woods.
I also recommend skipping that Nashville/New Orleans leg. You could do that on a long trip to Florida later.
With three drivers, I wouldn't stop the RV until I got to the Rockys.