If you're flying to Bilbao from the US or Canada, I think it's not a good idea to hop in a car and start driving 450 km. to Cognac. I'd be far too sleep-deprived and jet-lagged to be a safe driver under those conditions. If you're arriving from elsewhere in Europe, that's different.
I haven't been to southwestern France but did visit the Spanish Basque Country last year. I liked Bilbao a lot (preferred it to San Sebastian, in fact), but with so much territory to cover, I think you'll want to keep your time there to something less than 2 full days. Much will depend on how interested you are in art. There are two good museums; how much time do you anticipate spending in them? If very little, one day in Bilbao would be sufficient for the (largish) Casco Viejo and walking around the central area (pretty late-19th/early-20th century architecture).
San Sebastian also has a good museum. Its historic area is smaller, but again it's a pretty city just for wandering. The restaurant district is very lively (and touristy).
On the coast between those two major cities are many former fishing villages now with some tourist infrastructure. The ones I visited (all attractive) are Bermeo (near Bilbao) and Zumaia/Getaria/Zarautz (closer to SS). Several guidebooks mention Lekeitio, but I did not have time to go there.
Right at the French border is the larger town of Hondarribia, which is notable for having both a fishermen's district (restaurant area now) and an upper medieval quarter. It's large enough that one could spend a half-day there, or more with a meal.
The Basque countryside is green and very pretty; you'll enjoy your driving time, too.
