We made this drive last summer as part of an 8000 mile cross country road trip.
I would highly recommend entering Yellowstone through the Northeast Entrance in order to drive the Beartooth Highway. We traveled US 212 all the way from Minneapolis into Yellowstone. It was a beautiful drive and we were able to make interstate speeds once out of the Minneapolis/St. Paul area. Lodging wasn't a problem, we camped in the West Whitlock Recreation area on the Missouri river just west of Gettysburg, SD. This beautiful, clean, uncrowded camping area is part of the SD state park system. This was the week before Sturgis bike week, but we had no crowds or traffic. If traveling west on 212, there is no lodging after Gettysburg/West Whitlock until you get to Belle Fourche, as once you cross the Missouri, you will be on the Cheyenne River Sioux Reservation for much of the rest of SD.
In Montana, US 212 runs concurrent with I-90 in the Billings area. West of Billings, you can leave I-90, staying on 212 and go through Red Lodge, a really cool if a little over touristy town. From Red Lodge, 212 west will take you over the Beartooth pass. Many say this is the most spectacular drive in the US and I would agree!! This was a detour off our planned route and ended up being one of the highlights of our trip. On the west side of the Beartooth Highway we spent the night in Cooke City, just outside the NE entrance to Yellowstone. A great little town! We are planning to return to the Beartooth/Cooke City area on our trip this year.
Have a great trip whichever route you decide on!
Jeff