You realize that that's a 4-hour drive to Yellowstone, right? Even to the Tetons it's a very difficult 1-hour drive over Teton Pass.
Doesn't sound remotely practical to me, let alone with a baby.
Driggs to the West Yellowstone entrance is two hours, but its a fun drive (I had to stop for cows in the road). Then again I am not sure I would want to do it everyday. Then you are going to need to drive all around Yellowstone. Its not really all that practical. If its not too late, see if you can get a place in Big Sky MT. Its a little less than an hour from West Yellowstone and there is what to do in Big Sky itself. If you PM me, I can give you the number of place we stayed at, it cost us $200 on Airbnb (I used the gift cards from the Amex promo,so it netted me around $100 a night) for a nice three bedroom townhouse.