Rick's guide to Spain includes some coverage of the French Basque Country as well as the area south of the border. There's more than can be seen in a week, so you'll need to proritize based on your own interests. I assume you'll want to pick up and drop off the car in the same country to avoid the dreaded international drop charge. I don't think a single base would be practical. You might manage with two; three separate lodgings over the course of seven days would be a bit of a pain, but it could be done since you'll have a car.
Have you determined your departure point for the flight to Portugal? Is there something workable from Bilbao? There's also a smaller airport outside San Sebastian, but I don't know that it will have non-stop flights to Portugal; connections will drive up your travel time and probably your cost, and you should be sure you understand the ticketing situation. A compination of two flights on a budget airline may not be a true connection: The risk of a missed connection may all fall on you.
I prefer the larger, less touristy, Bilbao to San Sebastian, but plenty of travelers seem to like San Sebastian better. Other possible Spanish Basque destinations include Hondarribia, Zarautz/Getaria/Zumaia, Lekeitio, Bermeo, Vitoria-Gasteiz and Laguardia. In France I preferred the larger Bayonne to the cute St-Jean-de-Luz, and I like both a lot more than Biarritz. I didn't get to any of the small mountain villages.
I can't help with accommodations because I go for budget places in cities or large towns served by public transportation. I search for suitable ones on booking.com. The paradors in Spain are nice properties, many of them historic. There's one in Hondarribia and another in Argomaniz, a place I've never visited. I think you could stay in Hondarribia and visit San Sebastian from there (don't know about parking), but that would mean about a 3-hour round-trip to Bilbao.