I fly out of Seattle as well. We do have non-stops into Paris (Delta and AirFrance), but not Madrid. London Heathrow (British) is a possible transfer; also Amsterdam (Delta), Frankfurt (Lufthansa), and Reykjavik (Icelandic). It's easy to arrange an "open-jaw" ticket: just click "multi-city" on the airline ticketing website.
Personally, I recommend the 'open jaw' approach. Fly nonstop into Paris, take the airport shuttle-bus service from outside the baggage claim area, to the TGV from Gare Montparnasse to Bayonne, and the TER #62 local train to SJPdP.
Plan to walk your Camino to Santiago in about 35 days +/-. Add days for rest at SJPdP, enroute (Burgos, Leon, etc.) and a couple of days at Santiago if you possibly can. I usually allot 6 weeks more or less, when I do this route. That gives me a week or so to play with for slack time. If all else fails and Santiago does not entertain you, see Madrid before returning home. It is beautiful.
Depart Santiago either on an ALSA bus, or plane to Madrid. If you fly on AA (One World Alliance) you have a wide variety of routing options for returning home. Both Iberia and British Airlines are in the alliance with AA.
At Madrid, this means you fly into and out of Terminal 4, which is a big convenience and stress reliever. Your checked luggage is also transferred seamlessly. So, even though you cannot fly nonstop back to SEA-TAC, you can fly from Santiago to Madrid to (for example) Charlotte, Philadelphia, DFW or Chicago. From there, there are many flights back to SEA-TAC.
The other thing I found is that, if you leave on an early morning flight from Santiago (SCQ) you can make the connection to the noon (ish) flights from Spain back to the US. That is why staying in one terminal is a good thing.
Alternatively, you might be able to fly from Santiago to Barcelona and then nonstop back to SEA-TAC. I have not looked, but you might. Vueling air is a partner of Iberia and has its hub at Barcelona.
I appreciate that one might be able to shave a few dollars / euros by switching carriers or terminals, but WHY interject drama and uncertainty... I advise spending the extra money to have a more or less seamless journey.
Hope this helps.