We have made "loop" day trips into yellowstone from the east..we stay at Buffalo Bill State Park... we drive into the park, take a right at Fishing Bridge and go north and then north east out that gate and then take the Chief Joseph scenic highway back down to Cody and out to our camp site. It takes most of the day, but we take a picnic lunch etc.
so you can do it, and it's a good drive if you go either way...up chief joseph into the park and out the east gate etc...
but doing Teton Park from the east side is going to take longer I think.. lots of driving on 45mph roads.