I don't want to go down to Astoria but want to get on (connect) to the TransAm route as quickly as possible starting from Bellingham.
I believe that the quickest
way to connect to the TransAm is to go to Astoria, but you have ruled that out.
Let me assume that you what you want is not the quickest way to connect to the TransAm, but rather the quickest route to Yorktown using as much of the TransAm as feasible. I might also assume that you would temper quickest
with a nice route (e.g., the quickest would probably be to use I-5 and I-90 to get to Missoula, but that would not be an attractive route to me--but maybe it would to you). I might also assume that you want to do this entirely by bicycle.
So I might suggest you follow the Northern Tier to Sandpoint, ID and then take Idaho 200 and Montana 200 to Missoula. There are many factors and tradeoffs, and I don't know your view of these factors, so this might not be the right option for you.