Great advice above once you hit, say, Red Lodge. Like the others, it's not a good idea to pull anything major over Bear Tooth to Red Lodge.
I'd say we've not had a good experience overnighting in Wyoming, but Cody may offer some nice campgrounds. I don't know.
And as for the route from Balamer, we made a dozen trips from Bowie in the '90s so things may have changed. I'm sure you have that planned, but just in case. We'd take I 70 to Indianapolis, 625 miles for us, a bit less for you, the first day, then angle up to I-80 just before entering Iowa. If you think that portion of the trip was exciting, you'll love crossing Iowa, that's another 600+ to Omaha, Nebraska, continuing about 12,000 miles (really about 450 miles) into Cheyenne. Then start making choices, but you're close to great sights! Have fun and be safe.
