This is one time I don't recommend Fishing Bridge in Yellowstone for a 40' with 4 slides. The 40' isn't a problem. We always find a site. The 4 slides are what will be a problem. The only one for you in Yellowstone will be Mammoth which is no reservations but we always seem to find a spot there with our 40' (2 slides). However, you'd be just fine there with 4 slides if you stay outside the park nearby the previous night and pull into Mammoth early morning when folks are leaving. The sites are large there.
Having 4 slides and trying to stay in the national parks will not be easy because you have to postition those slides between trees and other obstacles. Both of our slides are on the same side so we have an easier time. I think it would be easier for you if you stay nearby, just outside the parks, in private campgrounds. Have a good trip. It's a favorite time of ours for those national parks.