Hope you have reservations already at Fishing Bridge, if not need to go to Plan B. If you can get by without power, would strongly recommend Bridge Bay. Same area but the RV sites are more spaced out and wildlife in the campground is just about a guarantee at least one of you days.
It will take most of the morning just to move from Colter Bay to Fishing Bridge. Not very far, but if you average 25mph between the two consider yourself lucky.
Think you left out a stop after Chamberlin as there is no way you are going to drive from Chamberlin SD to Cody WY. 600 miles plus going over the Bighorns and then a lot of 2 lane roads. We left Bridge Bay early to be at the East Entrance at 8am. Without much sight seeing, it was 5pm by the time we pulled in the Buffalo KOA.
The other days seem okay.
If you are going to Cotler Bay first, why don't you take the southern route across I80 coming up through Jackson Hole? You could even go through Indianapolis to avoid Chicago. If you are taking the northern route to see the Badlands, catch that on the way back.