From Seattle, the most efficient connections are probably none of the above. First suggestion would be fly to Paris, as you plan to do (you should be able to get a non-stop from Seattle) and then take the train to Hendaye from the Gare Montparnasse. From Hendaye you just walk out of the station, turn right and in a couple of hundred yards you'll cross the bridge and be in Irun. This will involve far fewer modes of transportation and changes then flying to Madrid, Barcelona or Bilbao, not to mention Biarritz. There's an airport bus directly from the CDG airport in Paris to Gare Montparnasse, and from there frequent trains to Hendaye.
Equally easy would be to fly to San Sebastian (if you can get good connecting flights from Seattle). As pointed out above, the San Sebastian airport is actually on the outskirts of Irun so you can walk into town from there, or start walking right from the airport, which is less than a mile from the Camino.
I rely on google flights for best flight connections. It shows, for example, a very efficient 1 stop flight from Seattle to San Sebastian on Lufthansa and a non-stop to Paris via Delta.
I'm pasting in a map from Windy Maps showing how close the "Gare" (station) in Hendaye is, and also the airport. (Look for the plane symbol below Hondarribia). The blue line running out of Irun is the Camino. The red line is an alternativo. You could spend your first night in Irun or Hondarribia (more charming than Irun), or else just start walking right away. There are some albergues and casa rurales on, or right off, the route several kilometers into the climb out of Irun.
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