Every route , I can think of is going to add 100s of miles to avoid , high passes.
You could stick to I-90 to Bozeman then south on 191 to West Yellowstone and day trip into the park, then back out on 287.
The whole park is 7,000' plus so you're going to have to do climbing to get there.
Like looking for a route Kamloops to Calgary , that doesn't go through the Rockies.
99DSDP 3884, Freightliner, XC, CAT 3126B, 300 HP /ALLISON 3060
2000 Caravan toad, Remco & Blue Ox.