Everyone has an opinion, so here's mine.
Beartooth Pass isn't all that steep, but it's long, and frankly a bit boring after a while. Coming from Red Lodge, after you've made the first turn into the mountain and climbed to the next switchback, you've seen the scenery. It's the ridge across the valley. You'll keep looking at it from increasing height for another 3-4,000 feet. The southern side of the pass (in Wyoming) has longer views and more varied scenery (lakes, wooded areas, and a waterfall). Coming downhill, it passes too quickly.
When I was there (mid-late August), the campgrounds outside Cooke City were posted "No Tent Camping" because of bear activity. NPS was considering closing Pebble Creek campground temporarily for the same reason.
The north side of Dunraven Pass between Tower and Canyon is the only area I'd have qualms about cycling on the east side of the park. If you hit it early, you could beat the south-bound traffic to the top. The rest of the roads on the east and south side can be cycled fairly easily, in the opinion of this road-hardened cyclist.
Coming out of Billings I'd regard as a transport stage, not much fun to be ridden as briskly as possible (or perhaps take a shuttle to Red Lodge). If I were doing this kind of trip, I'd plan on 2-3 nights at Red Lodge, 2-3 overlapping nights at a motel at Cooke City, and reserve one of the plywood shacks at Tower for a night. Watch the weather carefully and cancel one or two nights in Cooke City if the weather even mentions rain. Hit the Beartooth on a clear day and spend your nights inside in active grizzly territory. Sure, it'll cost more, but you don't have to get the $400/night Lake Hotel luxury suite, and I look at this lodging as money spent to make the trip safe and enjoyable.
Alternatively, you could hop over the ridge from Red Lodge to Belfry and head south on 120, then climb Chief Joseph Highway to meet up with 212 on the other side of Beartooth Pass (plan on a night somewhere in there, either an NFS campground or perhaps one of the ranches in the Shoshone valley). Chief Joseph (196?) is simply gorgeous.
If you haven't seen Old Faithful and the geothermal area below that, it'd be worth an extra day. Personally I'd stay off the roads on the northwest corner of the park -- go over from West Thumb and return the same way before heading south.
I'd stay on the west side of the Snake River (inside Grand Tetons NP) as far south as possible before arriving at Jackson. US 26 would be less pleasant to drive, what with the rented RVs being returned, the trucks, and all the other park visitors rushing back to catch their flight home. You may have two of the three inside the park, but there's something of an understanding that you take the highway if you're in a hurry.
