Returning to Biarritz airport?

There are several major towns in Spain that have bus service connecting to Biarritz through Bilbao and San Sebastian. There is nothing direct due to the issues with the border crossings. Two and half weeks out of SJPdP should be able to get you as far as Leon. Check out the Renfe and ALSA timetables, I suspect you will find yourself spending a night in Bilbao or San Sebastian before you can catch a Pesa bus to Biarritz. Both of these town are worth a one or two day visit.
Beware about renting a car, there are normally large surcharges for dropping off a car in a country other than the one it was rented, therefore if you rent in Spain plan your drop off in Irun and walk across the border to Hendaye and take the train to Biarritz. In addition most the Autopistas in Spain are expensive toll roads and petrol is about 4x the price in the US. Car parking can be a big and expensive hassle as well.

Biarritz Airport has bus connections to Hendaye, Irun, and San Sebastian. Transports64, PESA, and ALSA.

SNCF operates trains from Gare de Biarritz to Irun/Hendaye and Chronoplus C bus connects from the train station to the airport.

Renfe train connects to Irun/Hendaye from most major centers on the camino.

Euskotren connects Bilbao and San Sebastian to Hendaye.

There are buses from Pamplona and Estella to San Sebastian. ALSA and Estellesa.

There are buses from Burgos and Logrono to Bilbao and buses from Bilbao to San Sebastian. La Union Y Burundesa and ALSA.

There are buses from Pamplona to Logrono and from Logrono to Burgos that stop at most of the villages on the camino. Estellesa and Autobuses Jiminez.