One thing nobody else has commented on is, what are you going to be riding? IMHO, riding a mountain bike for 4,200 miles so you could ride 200 miles off-road (making up the numbers here, but that's probably in the ballpark) would be foolish. You're going to wish you had something that would roll coming across Kansas (and 9 other states).
If you stay on the connector and go into Glacier, you'll have 6 miles of dirt road between Columbia Falls and Apgar, and that'll be plenty if you're on the road/touring bike. (We went up that way, came back on the paved road, and had one problem -- a flat tire.)
To sum up my recommendations: TransAm to Missoula, Great Parks to Apgar, Northern Tier to Anacortes, all on (alleged) roads.