I was in this area last August.
If you have the time and have already gone that far I'd go north from St Mary and go to Banff and Jasper.
Anyway, that road on the East side of Flathead Lake is nice. Very difficult to find camping.
Rising Sun is AMAZING. Campsite at Apgar lake in Glacier was nice, 10 minutes from East Glacier Village and free showers.
Beartooth Highway (212) from Red Lodge into Yellowstone also Amazing. There is camping just past Red Lodge on 212.
I chose to go to Canada, Glacier and Yellowstone in August and save Monument Valley for cooler weather. If I were you I'd head North rather than South in the heat.
The night before crossing Nevada I stayed in a hotel. Meaning I didn't have to pack up camp could get an earlier start. I left Utah early morning and was in Winnemucca for lunch.
Canada tips
Highway 3 through Fernie is nice. 93 from Kootney to Lake Louise is nice, 93 from Lake Louise to Jasper is on par with Road to the sun or Beartooth Highway. Jasper to Clearwater is Awesome too. Camping is more difficult to find in Canada. Don't under estimate the weather, even in August. I used both my heated grips and Hippo Hands and was glad I had them.
Glacier NP
IMG20190819134736 by
Norcalkid, on Flickr