Sounds like a great trip! We're doing the same trip 2 weeks after you, staying in Fishing Bridge for 2 nights then Rustic Wagon in W.Yellowstone for the next 5 days. Since we're staying for 3 nights in Cody, we'll be taking Hwy 16 out of Buffalo. I found this info from an earlier post on this forum dated 6/30/16.Titled "Devil's Tower to Cody Wy Hwy 14 - We did the route a couple of weeks ago and opted to take 16 across the southern route instead because I get nervous on mountain roads. It was a beautiful ride. We had no white knuckle moments. We especially enjoyed stopping and playing in the snow up at the top of the pass. Bottom line, if you're nervous at all, take 16 for a smooth, beautiful ride..
Looks like going the I-90 to Hwy 191 route is a quite a bit further. Anyway, have a wonderful time and watch out for them bears.