With such a short time window, homework will allow you to narrow down your must see areas in Yellowstone and Grand Teton. I would definitely recommend Fishing Bridge unless your particular interests override that. As has been said above, the distances just within Yellowstone are substantial, and I would not recommend trying to drive to from Teton daily. Drive once and spend an hour or two checking in and setting up. You won't regret it.
I would also recommend Colter Bay RV camp for Grand Teton. The non-RV camp ground there has a number of sites that have electric hookup and there is a dump station there, but I just found the RV camps made it easier to come and go with a minimum of fuss. And I am a pretty die hard non-RV Park camper.
There are sites still available in both for Sept., but size selection is already limited.
Enjoy, and yes, you will be back after you get a taste of what these beautiful parks have to offer, even with all of the people there.
BTW, I highly recommend booking a dinner at either the Old Faithful Lodge dining room or the Lake Yellowstone Hotel dining room. You can make dinner reservations up to 60 days in advance.
A Happy Hour cocktail and some lite food at the Blue Heron Lounge at Jackson Lake Lodge in Grand Teton is a very nice break between the days and evening events.