We stayed at Fishing Bridge around 6 years ago when we first purchased our fifth wheel. Our reservation was for the last week the park was open and ended on September 20th. The second to the last night we had considerable rain and snow mix at Fishing Bridge, but nothing stuck. The East Entrance was closed with a reported 4 feet of snow on the pass. We bugged out a day early as I wasn't prepared for winter driving.
We stayed at the Tetons prior and had a similar decision regarding Teton Pass. We decided to circumvent it.
We were lucky in that it only rained at night and we thoroughly enjoyed the park. We did hit some snow on Dunraven pass which added to the beauty. Driving and parking in the park was easy, however the main attractions were semi-crowded due to the tour buses. I wouldn't let the September time frame deter you, just come prepared more-so than what I did.