It's been 20 years since I rode through the park, so maybe things have changed for the better road-wise, but I can think of better places to ride than Teton Park. From the south of Yellowstone to Jenny Lake was nerve racking traffic-wise.
And note that you may not practice dispersed camping in the park. You may only camp in established campgrounds.
You're in Bozeman. Why not start out with something more local? Three Forks is a scant, flat 31 miles away. From there it's a short hop to L&C Caverns. From there, the Bike Camp at Twin Bridges is not a hard ride.
Or if you want to get into the woods on fairly easy gravel park in Philipsburg, ride over the hill and down to Rock Creek Rd. and hang a right. There are at least five U.S.F.S. campgrounds with water between there and I-90. 30 miles of gravel and 10 miles of paved road. Beautiful road along a blue ribbon trout stream that's know for its moose sightings along with its fish. Just before the Interstate there is a private campground with a decent restaurant:
https://ekstromstagestation.com/Would make a nice out and back excursion with only one hill in each direction. If you spent the night before at the Inn/campground in Philipsburg, the owner might allow you to leave you car at the place. Unless she's sold the place, she's a hoot.
http://www.theinn-philipsburg.com/