I, too, would take the longer connection - even if the weather is bad, and Seattle can certainly suffer from rain, a trip downtown on the light rail would take 45 minutes or so from exiting the airport and Pike Place Market is always good to walk round - more on the lower floors where there are some hidden gems of shops, but my favourite is a wine tasting retail unit which overlooks the water. Beneath the market, at street level on Alaskan Way, is Copperworks Distillery Company which again offers a relaxing tasting experience with views over Puget Sound.

A hidden gem as well, in my eyes, is the nearby Library where they offer self-guided tours - the design is interesting and the ephemera content even more so.

If you really don't want to leave the airport, AS's lounge at SEA (the one in the North Satellite, don't use the small D concourse one) has possibly the best lounge bar in North America and the most helpful staff who are knowledgeable about what they're serving. It'd be a decent hangout until you need to go to the BA lounge.

The Museum of Flight is also about 15 minutes from SEATAC - traffic-wise, this is a much less risky trip back compared to going a lot further north to the Boeing factory at Everett.
GUIDE to touring Boeing plant, Flight museums in the Seattle area

2h20 on separate tickets is, as others have opined, a recipe for disaster - particularly when there so much more opportunity with the longer connection.