Several years ago we drove our then 15 to 18 year old Bounder over the Buckhorns to Cody and we were pulling the Suzi Sidekick. We came from Mt Rushmore and we stayed at Crooked Creek Campground which is about a 1/4 mile closer to Hill City than Rafter J. It's a small campground but less expensive than RJ and easier to get in at the last moment. We couldn't get reservations at RJ so had to look elsewhere. Doing it again I'd stay a Custer.
Being self contained we didn't need full hookups at Yellowstone and Fishing Bridge was full. The DW got us a space by calling the Park rather than the reservation website. The gal she talked to got us three nights in the same site when the website said there were no spaces available. We wanted to stay an additional day and for that we did have to move to another campground.
The journals we keep are in the garage in boxes while we refurbish our S&B so I can't give the names of the campgrounds, all I know for sure is it wasn't Fishing Bridge and the access road around the loop was very narrow and getting in and out caused a traffic jam no matter what day or time of day.
Have fun and let us know how it went.