Welcome to the forums. Do not let bgun's comment bother you as your rig looks like it's about the right size to me. Way too many weight police on here. No need for reservations in Sept but I'd high tail it to Yellowstone as the later in Sept the colder it can get... snow likely.
Only you can decide how many miles/day you are comfortable driving and the reason I recommend no reservations. X2 on
Baker's Hole USFS CG just north of West Yellowstone. First come so the earlier the better to arrive... before 11AM and even if you cannot get an electric site the first day you will be able to get one the next day. There is a dump and water as you enter on the left and the camp host will be on the right on the exit road.
I like this
site for finding CGs as it uses a map and you can select the type of sites you want. I would do 5-7 days in Yellowstone and 2-3 in the Tetons. Many stay at
Colter Bay CG or if you want FHU...
Colter Bay RV park.