Fast train to Bayonne from Paris & use Bayonne as your home base to explore Pays Basque (on both sides of the border).
After a 5+/- hour fast train from Paris you're in Bayonne and can then strike out for San Sebastian or Bilbao, Spain; or closer to 'home', St. Jean de Luz and/or Biarritz in France, or head east into the real Basque country. If 'twas me, I'd rent a car in Bayonne and get into the real Pays Basque just a few miles inland (as opposed to the ersatz Basque country that Rick speaks of right on the coast).
Some small towns/villages to think of visiting: Ustaritz, Espelette, Ainhoa (France); Dancharia, Elizondo, Santesteban (Spain). Do a loop from Bayonne through the towns above (did it in September), and you'll easily have a grand time in less than 7 days.
Turn the car back in at Bayonne, and off for a 5+/- hour train trip back to Paris.