Here:
https://ridewithgps.com/routes/29469149?beta=falseNo Interstate riding at all. I've most of this mileage at one time or another, and most of what I have done was ridden at least once since 2011.
The parts I have not done: MT 84 between Bozeman and Norris and MT 569 between MT 43 and Crackerville Rd.
Th Byway to Wise River is a terrific ride, especially the alpine meadow section.
At mile 240.5 the road turns to dirt for a few miles. It's benign. 32c or better tires should be fine. There is actually a paved trail in that area that will get you close to Butte, but it's not yet on the map. A local I ran into showed me it. The goal is to have a single trail between Anaconda and Butte. Do some research. Maybe it's finished.
At mile 247.9 you pick up the BA&P Trail that takes you up to the Old Town section of Butte, which is the historic district with cool old architecture. It also has the far less traffic than the sprawl to the south on Harrison Ave., etc. Routed you passed the Hotel Finlen. IMO, It's the place to stay in Butte. Crashed their three times. Get a room in the motor lodge portion. It's not that expensive. The Butte KOA is right next to I-90 and doesn't seem very nice.) Ask for a room on the first floor so you can easily take your bile inside. Treat your self to a steak at Casagranda's. Soup, salad and dessert included.
Pipestone Pass is pretty easy heading east. Then you have a nice, mostly long downhill to Whitehall. If you camp at L&C Caverns State Park be advised that the campground is at road level but the caverns themselves are up a steep, multi-mile climb, and there is no shuttle. You might be able to hitch a ride from fellow campers or day visitors.
If you don't want to loop around to Anaconda to get to Butte you can continue east down the canyon on MT 43. take frontage roads north for some of the way then ride a relatively short stretch of I-15 towards Butte. Take a look at Google Maps Street View of that section of I-15. Not exactly what I would call busy. I have hung out under the overpass on the road between Melrose and Twin Bridges to take shade breaks and put on sunscreen. Sometimes many minutes passed without a vehicle going by.
And try to stay at the Bike Camp in Twin Bridges. Great facility for cyclists.
Let me know if you have any questions. I have detailed files.