I've done the Sawtooths a couple of times -
They make a nice west-to-east route across Idaho - although Stanley can be COLD even in July.https://www.crazyguyonabike.com/doc/page/?o=tS&page_id=421648&v=1c
I cut off the TransAm at Austin Jct. - staying on US 26 - with some killer downhills and tailwinds.
You can follow the Payette River all the way from Ontario to Banner Summit.
Great hot springs along the road between Garden Valley and Lowman.
(Services are extremely limited - the store/cafe at Lowman is closed, but there are a few lodges.)
I've gone thru Challis and Salmon on my trips - not Sun Valley - all beautiful.
Once you get out into the Snake River Plains it can get hot, esp. around Craters of the Moon.
I prefer ID 22/33 to Driggs over US 20/26 to Alpine. Teton Pass is tough,
but there's almost as much climbing along Palisades Res. with narrow winding road.
Grand Teton is spectacular with a great bike trail. Jenny Lake has hiker/bikers camp sites.
They are making it harder and harder for for cyclists at Yellowstone's South Entrance.
I find that climbing up the plateau is easiest late in the day.
Traffic is lighter, you are shaded from the western sun, and hiker/biker sites are at Lewis Lake.
(I hope you are doing the big loop from West Thumb to O.F. to Norris to Canyon to Lake.)
The ride from Pahaska Tepee to Wapiti is truly spectacular along the Shoshone River
Be aware that there is no tent camping from Bridge Bay in YNP for the next 40+ miles east.
And you should hit the peak of wildflower season in the Bighorns.
Oh, yes - South Dakota.
Have you considered heading down to Spearfish and up Spearfish Canyon - super-duper!
Then you can hit touristy Deadwood and Nemo Rd into Rapid City.
Then SD 44 takes you to Badlands NP - which is really nice in early summer - brutal later.
If you are willing to do a little hardpack dirt - Sage Creek Rd from Scenic lets you ride the entire park.
(Or less dirt - 20 mi east on SD 44 to Conata Rd, 9 mi dirt)
Then a combination of US 14, SD 34, and maybe the back route I mentioned above?
And as for the Cheyenne River Res. - conditions are deplorable - and there's no excuse.
The two counties are roughly 80% Native Am and 20% white and many ways like apartheid South Africa.
Many communities lack water and sewer - substance abuse is epidemic. And horrible wrecks.
(Many it's random - but I've seen too many on my tours thru Indian country.)
The people I have met on reservations have been wonderful - but as a non-native, it's not easy.
Best - J
Pic - Badlands in June