Some real different responses from our experiences there in our 3 visits over the past 7 years, always the later part of Aug. into the last of Sept.
First - Fishing Bridge usually closes down after Labor Day weekend. HOWEVER, the other campgrounds will be open, but they don't have hookups. They do have fresh water fills and dump stations however. We have stayed at Madison, which is our favorite and right next to the river. It is also the closest to West Yellowstone. Mamoth is at the north end and you have long drives to Old Faithful and the lake.
Second - Yes, weather MIGHT get iffy into Sept., but we have been there during the 3rd. week of Sept. and still in the mid to upper 70's+. The week before, it was still in the low 80's. Yes, it can vary, but generally no problem until the latter part of Sept. Yes, you may get snow and cold nights, but generally the snow doesn't stick around during the day.
Third - I really don't view West Yellowstone as the "middle of anything"!! Especially if you are there to see the real wonders and sights of Yellowstone Park!! WY is a "tourist trap" that gives the illusion of Yellowstone!! But, different strokes for different folks! :B It is probably the only place to rent a car however.
I would certainly make time to drive the Bear Tooth Scenic Drive outside the northeastern entrance to the park. One of the most spectacular and scenic mountain drives in the lower 48.
Have a great trip and enjoy one of the wonders of our country.