If you really want to do Arizona and that area then you might be best to fly somewhere closer and do a tour from there. Las Vegas might be a good location or Salt Lake City. I think that you could see a lot from man of the Western cities and you would not have to spend so much time driving there. As someone else mentioned, the saving on gas would probably pay for flights.
If adverse to flying then you could consider driving your own car a bit closer. For example, you could maybe drive to St Louis and rent an RV there. That way you could cover more ground faster (particularly if you and your other half shared the driving, even overnight) and that would give you more time for the rest of your trip.
I can say that there is plenty to see and do in South Dakota as mentioned above if Mt Rushmore was to be your number one destination after the Grand Canyon.
Be aware of the Sturgis Rally as the prices shoot up and it is very very busy. The RV Park we stated at was quadrupling their nightly rate.
10 Days and Nights of Riding, Food, and Music. 500,000 riders, concert fanatics, and street-food enjoyers attend the Sturgis Motorcycle Rally every year.
www.sturgismotorcyclerally.com
As well as Mt Rushmore I believe the Borglum Story in Keystone is worth a visit. It shows as temporarily closed - maybe because of Covid. Please don't be offended - we were rather underwhelmed by Mt Rushmore and much preferred the Crazyhorse Monument, although we didn't get there until dark and missed the museum.
Here are some of my recommendations for that area:
Devil's Tower - hike around the base (around an hour's drive from Spearfish)
Crazyhorse National Monument - really great museum and interesting story of the monument. I was really glad we went. We spent several hours there.
Deadwood - gunfights in the street, Trial of Jack McCall, old cemetery, couple of old bars (Buffalo Bill original bar is downstairs)
Lead - Sanford Homestake Gold Mining Museum is interesting
Spearfish Canyon - check if suitable for RV, couple of nice hikes
Custer State Park - beautiful place and worth a few days at least, believe there are lots of bison there.
Note that the Eye of the Needle Highway is not suitable for an RV.
Sylvan Lake is really lovely.
Black Hills Burger & Bun was a great burger joint in Custer
Wall Drug - a huge shop that seems popular. Watch for the adverts on the highway as you approach, some were very amusing.
Badlands - really lovely with lots of hikes. Cedar Pass Campground looked really nice but I believe there is lots of boondocking around. Watch out for prairie dogs at Prairie Town, and if you watch closely you might be lucky enough to see Burrowing Owls nesting there too.
Other places around the area that we have not been to but heard of:
Wind Cave
Minuteman Missile Museum
Tatanka - our campground neighbours liked it - story of the bison