Chances of getting a site inside Yellowstone are slim but worth a try. With your 24'er you might get lucky in one of the dry camping CGs. Much better to be inside the park to start & finish the day than have to commute. Check the distances. It is a BIG park. Combine that with bison jams & much time is needed to get around.
My calculation, staying in the park, is 3 nights minimum. Arrival & hit a visitor center day, 1 loop day, 2nd loop day. We are not hikers but we are not the people that you see drive into a scenic area & get out of the car for 5 minutes, if they get out at all, then dash off to the next one & claim to see Yellowstone in one day.
Teton is also well worth two full days for the non hiker. The Teton dry CGs are non reservable. Get to one before 12 or early afternoon & there should be no problem. We were at Gros Ventre CG over the July 4th holiday last year. It was full only 1 night. Teton is also a BIG park so outside CGs are a drive away.