I think it is doable. Haul bananas to Vegas and have DW fly and meet you there.
Then Willams and take the train to see Grand canyon for the day. 2 or 3 nights.
Then ZION and Bryce for maybe 3 nights each.
Then onto Tetons for a few nights and then to Yellowstone for maybe 4 nights. Lower loop one day, upper loop second day. Out the east entrance to Rushmore.
One night en-route and two nights near Rushmore. Stop in Cody WY on the way through for a few hours to see Buffalo Bill museum.
After a full day at Rushmore and a nights' sleep you high-tail it back home.
Get your calendar and mark the days. Get reservations where you can especially inside Yellowstone. Going to be tight so plan it out and know where to cut and run if needed.
I would skip Vegas/Hoover before I skipped Yellowstone.