If you are not checking a bag and you already have your tickets, the only "check-in" time that applies to you is the time the conductor yells "All Aboard". You can literally arrive at the station a minute before departure and still catch the train. There is no need to "check-in" with a station agent, you can go directly to the train platform if you're running late.

Amtrak has a system-wide "30-minute check-in" time, but the website makes clear it's only a suggestion. Nonetheless, it's a good suggestion for those unfamiliar with a station layout, or larger stations that require stairs/elevators/etc to reach the platform. Not much of that applies in Bellingham. As long as your flight is reasonably on-time, you'll have no problem catching your train.