We did a monthlong trip May/June of 2020 from Tucson. September would probably be decent for weather. If you wait until all the kids are back in school {a good thing} you will be pushing the weather envelope as fall/winter comes early. be prepared for all sorts of weather, we had snow flurries and temps in the low 30's all day on June 4th... just sayin.
We have friends in Angel Fire NM so started our trip at their place for a few days and then went north through Colorado to Wyoming. Crossed WY from south east to northwest stopping in Cody. The Wild west museum there is amazing g and a must see. This brought us into the park at the east entrance and we had reservations just outside of the west entrance which was ideal. I recommended Grizzly RV in West Yellowstone, a nice RV park just half a mile from the west entrance to the park.
We got a great preview of the park coming in from Cody and when your daily visits start at the west entrance you can beat a lot of traffic as there are several road options as you enter. If you come in from the south you will sit in line for an hour or so just to get in.
Start your days as early as you can to help avoid the worst of the crowds and go back late in the day if you have not had enough. The best wildlife viewing is early morning/evenings. Do NOT miss the Grand canyon of Yellowstone, probably the most spectacular area of the park.
When you head home allow a couple of days at least for Grand Teton NP and the Jackson area. Heading southwest from Jackson follow the Snake River Canyon down into Idaho, one of the most spectacular roads I've ever had the pleasure of driving.
I hope you have half as much fun as we did, the entire trip was awesome. Good luck.
Here is a shot of one of the falls in the Grand Canyon section of Yellowstone:
This Was taken in Grand Teton NP. We tow the bike in our cargo trailer behind our 24' Class C. If you want the ultimate National Park experience {in any NP} see them from the back of a bike... but I digress:
:B