I would definitely recommend the early September date. Even then, the nights get cool. By the end of the month, you might call the nights "cold"! (We even had snow around September 10-12 when we flew out...after a five-hour snow delay!)
The early September weather is beautiful. Another reason is that some of the services in Yellowstone and the Tetons (which you should definitely visit) are curtailed shortly after Labor Day.

I have been in Yellowstone between the 5th and 20th. I think that your chances of beautiful weather are better early in September. The times are a changing out there by the advent of fall.
The problem with mid September is that most of the summer ranger staff has departed and the interpretive program is reduced. On the other hand, the crowds are smaller.

Also lodges begin to close and it becomes harder to find a place to stay overnight in the park. If I had to make a choice, I personally would go about September 8.

A lot of the jobs are summer jobs for college students and they leave for school by Labor Day. However, you'll find a few more services in early- rather than late-September. There is only a slight difference in the weather between early- and late-September. Those Indian Summer days are fabulous and you should enjoy them. It's one of my favorite times of the year. Plan for crisp evenings and mornings.

I'd recommend the late September trip! We went last year to Yellowstone and the Tetons on September 18th - 27th - the timing was exellent: less crowds, the elk rut season was wonderful to experience, and the wildlife is quite plentiful and active in the fall. It was "light jacket weather", very comfortable, with only a brief shower on the day of our arrival. The so-called "negatives" of a late trip [ranger-led programs are done for the season, some lodging is closed, etc] were unimportant to us and more than
offset by the wonderful fall experience...and oh, those wonderful fall colors! Don't let the late date deter you....it is the best time of year in Yellowstone!