I recommend you do some weather and National Park research. Spring is wet and cold and snowy in the NW. Good riding starts in June. We have nice days or patches of good weather before that, but it doesn't really dry up till mid June.
Yellowstone and Glacier have roads closed for winter. Going to the Sun upens mid June to early July. Yellowstone a bit sooner, maybe May. Unsure about Beartooth.
If the weather co=operates June is a fantastic month for green grass, fresh weather and glorious sinshine.
Maybe start in the south and come north as the weather dictates.
4 months sounds grand.