Stick with BA. Despite being on the west coast the flight time is such that going via anywhere, with the possible exception of Vancouver or Calgary, adds so much time it's simply not worth it unless you make a benefit out of necessity and stop at your transfer point. And... because there aren't that many international flights to Seattle (even allowing for the Pacific) customs/immigration is a breeze (or it was this summer).
A couple of thoughts. International arrivals are processed at a satellite. Once you exit customs you are transported to the main terminal by a train (similar to Heathrow T5). There is a parallel baggage conveyor that they encourage you to use (and I did) but I don't believe you HAVE to. The other is look at WTP out and Club back.