I have to agree to avoid highway 14, steep, winding narrow road, especially from Greybull. And as mentioned, the KOA is very nice, the owner is a great guy!
The route from I-90 to West Yellowstone, 191 from Belgrade south is not too steep, but tight, narrow curves from Gallatin Gateway to Big Sky.
212 through Red Lodge has a lot of steep inclines, and sharp corners, and switchbacks,
89 through Gardiner probably has the least inclines and wider roads, but it does go through the park, so wildlife traffic jams, as well as general heavy traffic flow should be taken into consideration.
Each route has their challenges, but with a 1500 and a 30’ trailer I would not take 212, that will be tough on your truck and you.
Also depending on the snow season this winter, Hwy 14 and Hwy 212 could still be closed into early summer, both are closed in the winter.
2011 Dodge Ram 2500 Cummins
2012 Starcraft Autumn Ridge 297 BHS