Taking I 00 all the way to Billings Is much longer than crossing the Bighorns, and going through Cody. You had one suggestion to take US 14 over the mountains, and on through Cody. This route is quite scenic, but it is steep. My own favorite route is US14 up the Bighorns, and then US 14A down. IMO, this is the most scenic route, but it is also the steepest.
The easiest way to cross is US 16. It will also take you to Cody. where you can pick up US 14 ti Yellowstone. This is not a difficult drive. It is not as steep as US 14, and is probably also a little bit shorter.
I would hesitate to try to do it in one day. If you can manage one night without hookups, there are plenty of national forest campgrounds in the mountains, on 16, 14, and 14A. There are also some commercial campgrounds with hookups. One is on 14, a few miles before the junction with 14A. There is another on 14A, about half a mile after the junction. If you use that one, you probably want to go back to 14 when you leave.
This sign is a good indication of the roads: