Long days no matter what; if you want to take US 212 you could also camp at/near Hardin MT in place of Buffalo WY. A bit longer drive from Mitchell, but will make the drive into/through Yellowstone the next day a bit less long than it would be from Buffalo.
If you do decide to stay on 90 and stay in Buffalo, when you head out the next day avoid US14 and 14A, and be aware that US16 will also be lots of climbing, winding roads, descending, etc. over the Bighorn range - much slower going than the interstate. It may be shorter but probably won't save you any time.
From 90 near Laurel MT you can take US 212 and then US 310 to SR 72 (which turns into SR 120 in Wyoming) to drop down to Cody - not too hilly, still plenty scenic, and it may save you a bit of time over towing through the park from the north entrance. From Cody head west into Yellowstone, go in the east entrance and out the west. Towing through the park is slow, as is all driving in the park - traffic, animals, animal jams, and the 45 mph speed limit all make it at least a 2-hour drive from east to west across the park. From the north entrance, you'll encounter MAJOR road construction from Mammoth to Norris, which will slow you down even further, hence my recommendation to go in from Cody.