Two suggestions for smaller streams with cutthroat and easy access close to West Yellowstone would be the upper reaches of the Gallatin, going north on Rt 191 from West Yellowstone, fishing above Big Sky (parts are in the Park, and parts are out of the Park, where you’d potentially need both a MT and a Park fishing License depending on which sections of it you’d fish). Also, there’s the Gibbon River, probably above the Gibbon Falls.
Driving in the Park can be a nightmare, so both of these would minimize your driving time going into the Park from West Yellowstone, however, if you don’t mind driving farther, there’s also Soda Butte and Slough Creek out in the Lamar Valley.