You should have no problems.
We have done a number of similar trips, 2-up on a 2000 LT:
last summer was farther north in WYO (from I90, thru Cody, to BearTooth Pass, and up into Red Lodge Mont);
+maybe 3 yrs ago we wandered the central part (Lander, Riverton, Thermopolis).
I agree with George S (below) - once you get to 1/2 tank, refill at the next station you see; it could be another 80-100 miles before you see one.
As for the painted canyon, I think you'll find it about 30 miles East of Montana, just into ND at Theodore Roosevelt Nat'l Park near Medora right on I94. We came back through there last year.
When in Boise, ask a local how to get to Bogus Basin, just north of town.
The road leads to a ski area. Great road, not fast, 2nd gear for about 15 miles - lots of fun.
Our trip last year was US12-Lolo pass, south to Boise (to turn around) then NE thru Lowman to Salmon, where there is a great B&B in a log building.
I forget the name, and it's not easy to get to but neat people run it.
HAe a great trip