We did basically what you are looking at last year. From Missoula to the Trail of the Coeur d'Alenes, as far as I could tell there's really no great way. We used I-90 from Mullan, ID (east end of the trail) to Missoula for the most part. Where we could get off the interstate and use parallel roads we did, but there are sections where it didn't seem possible, like the 30 odd miles from Lookout Pass to St. Regis. There is the old rail-grade that goes over Lookout Pass that we could have possibly used, but since no one could give us a straight answer as to what condition it was in (we knew it was unpaved, but didn't know how graded it would be) we just took I-90 over the pass. You could use MT 135/MT 200/US 93 to get from Missoula to St. Regis, but it would be longer and you would still need to use I-90 from St. Regis to Mullan.
Westward from the Trail of the Coeur d'Alenes, the route we took departed the trail at Harrison, ID, headed north on ID 97 on the east shore of Coeur d'Alene lake to where it intersected I-90, got on the freeway for a couple miles, and then got off to follow the Centennial Trail through Coeur d'Alene city into Spokane. From Spokane northward we used US 395 to connect to the Northern Tier/WA 20 at Colville. (Note: when we did this trip we were travelling west-east.)
Choosing the Colville-Spokane-Coeur d'Alene-Missoula route required us to make some compromises, like quite a bit of freeway riding in Montana. We found it tolerable, if not perfect riding. Thankfully the scenery was beautiful and the traffic not so bad. Others might not like it.