If you're on one ticket, this is very easy and 2:49 is a ton of time. Even if it's separate tickets, Alaska
policy still allows through checking to the partner airline (unlike, for example, American Airlines). In fact, they'll even base the checked bag fee on the allowance on your international ticket, so you shouldn't have to pay the Alaska bag fee (if one normally applies to you).
Travel that includes multiple airlines or tickets
If your travel includes multiple tickets, and/or travel on more than one airline, your baggage fees and rules may be determined by the other airline. Please check your ticket, or call Alaska Airlines Reservations at
1-800-252-7522 to help determine which airline rules and fees apply to your journey.
When making international connections on a separate ticket to other airlines within twelve (12) hours of arriving in the connection city, you may follow the baggage allowance and weight restrictions of the international airline - provided a ticketing and baggage agreement is in place between Alaska Airlines and the other carrier.
Given that, I'd be comfortable with far less than 2:49 minutes in Seattle. You may have to go to the BA gate to get your BA boarding pass (if you can't get it online), but you won't have to leave the secure area. You'll almost certainly need to go to a human rather than the kiosk to check the bag at GEG if you're on separate tickets, so I'd allow a few minutes extra time at GEG. I haven't done this (check from an AS outstation to a long haul ticket on a separate carrier) in several years, but when I last did (a decade ago) it was totally routine and the Alaska/Horizon check-in staff were efficient. Have a printout of your connecting long haul ticket.