The advantage of boxing the bike is it goes as checked baggage. You won't need to transfer yourself for any other connections. But sounds like you are doing it in one shot. I would go with the Cascades idea.
IMHO, unless you will be camping or hope to grab a room each night then it is a mistake to take the train to Portland. It can be as much as a day late. May screw up an itinerary.
Also the EB is better eastbound. You go through Glacier during the day eastbound.
An option might be to take the Southwest Chief to Chicago from LA after you are finished and take he EB home.
Or take the Starlight back to Seattle from LA. Then take the EB home. Fly out to Portland.