I am not current on BNSF dispatching for the Seattle Area, however since there are no Sounder stops between King St. Station & Tukwila, I'd echo the comments above that they can & do use any track that suits their needs at the time. The biggest change I'm aware of is that U.P. trains will often stay on the West Main ( track 1 ? ) between Black River & Argo; in the past N/B's would be crossed over from West to East @ Black River, just to be crossed over again @ Argo / Lucille St.
I recall there is ( was ? ) a Best Western Inn at Pioneer Square. Also several hotels in the area of Seattle Convention Center. If you want to stay near or at SeaTac Airport, you should find your way from King St. Station to the Sound Transit Link light rail station underneath Union Station across the street to the East. Grab the next Southbound, which will ( in a round - about way ) take you to the stop at Tukwila International Blvd. station, and / or SeaTac Airport station. Lotsa hotels within walking distance along Int'l Blvd.
Trolls can now enter negative comments regarding public safety while riding Seattle area transit.