September can be an absolute beautiful month for Yellowstone. We've been there at that time a few times and never had snow or heavy freezes. Of course, every year is different. I'd try to be out of the area by the 3rd week of September and begin heading south.
You don't have to winterize during your trip. If it freezes at night it will warm up during the day. Fill your water tank and skip the hose outside, especially if you plan to leave the next morning. The hose will be frozen. Same for your sewer hose... keep it stored until you need to dump. Of course, at Madison you'll have no hookups so you don't have to worry about it.
I hope you were considering seeing Grand Teton Nat'l Park also. They're right next door. Gros Ventre campground in the park is a nice place to stay. There are a few sites with electric but you have to ask for them. The campground has 300 sites and none are reservable and it rarely fills, especially in the Fall. There's a dump station and water fill. You'll be close to Jackson for major grocery shopping just south of town and for general small-town looking at shops.
A good route would be to work yourself along I-80 at some point and then exit at Rock Springs, Wyoming. Take Hwy 191 north into Jackson, WY and drive through (and hopefully spend a couple nights) in Grand Teton. Then continue north through Grand Teton and enter Yellowstone from the south.
When you leave Yellowstone, definitely head into Utah and visit the awesome national parks there. Arches, Canyonlands, Capitol Reef, Bryce, Zion, Monument Valley (not a national park but nice) and perhaps even getting into Arizona and the Grand Canyon before heading east on a more southerly route to your home. All of these other parks have great weather at least through mid-October.
Can you leave home the last week of August? That would give you a leisurely drive to reach Yellowstone. 300 miles in a RV is tops for us. That's about 5-6 hours, pull in and relax a while. You'll be more refreshed for those daily drives it you don't push it - IF you have the time. It sounds like you might be retired so enjoy the trip and don't make it a rushed one! How often will you do it?
OOPS! Just re-read your post and you said the LAST week of September. I was thinking the first week. If you can change your plans I really think you'd be happier more toward the first part of September. You might luck out with the weather the end of Sept. but you will likely have some snow and lot colder nights and perhaps days. Just a suggestion.
Full-Timed for 16 Years
.... Back in S&B Again
Traveled 8 yr in a 40' 2004 Newmar Dutch Star Motorhome
& 8 yr in a 33' Travel Supreme 5th Wheel