Instead of heading north to Minnesota - cross Iowa, northern Nebraska, and southern South Dakota.
Then continue across northern Wyoming to Yellowstone and up US 89 to Glacier.
From there you can reconnect with the Northern Tier or do whatever.
The Iowa DOT bike map gives you great info for county roads in the state.
Combine back roads thru Iowa City and Des Moines with paved trails up to Sioux City.
Hwy 12 is a superb route in Nebraska - Hwy 44 is good in SD until you get close to Rapid City.
The area around Rushmore is a zoo. Other parts of the Black Hills are way nicer.