Presumably, since you mentioned Bike Week at Sturgis, you'll be there some time between August 3 and 12? Then a week in Yellowstone, driving south to Kanab for several days...sounds like that would put you at the Grand Canyon in late August.
Having just done the GC, both rims, in October 2017 I would strongly recommend the North Rim. It will be cooler, less crowded, and quieter there. The campground inside the park is no hookups, so keep that in mind, depending on your needs. There's a couple of FHU campgrounds at Jacob Lake, which is about 45 miles north of the North Rim Visitors Center. Since your wife has limitations there are still points of interest to see, that would not overly tax her health, and to which you can drive for photo ops.
If you decide to do the South Rim, check out Trailer Village inside the park. FHU sites are available (for A/C) and there's a shuttle that can take you everywhere you want to go. There's 3 very distinct areas at the South Rim, each with appeal depending on your interests. There's also museums and lots of information on the formation of the Canyon and it's historical development.
We were there for a week (family hiked down to the Colorado River from north to south) and there was plenty to see and do while there. You will not be bored.
ETA: Elevations. Will impact the weather you experience. It will be generally cooler. You don't indicate your starting point, which could be important. Sturgis is around 3400 feet. Yellowstone is around 8000 feet average. Kanab is 4200 feet. North Rim of the Grand Canyon is 8300 feet and South Rim is 6800 feet. Going the way you plan, starting in Sturgis, is actually a good idea, since you and your wife can acclimate to altitude gradually, which is the recommended way to do it. And, with the exception of some passes in the Yellowstone area, you'll be under 9000 feet the entire trip. Certainly doable for folks with health issues, if proper precautions are taken.