How to get to Biarritz from seville

Fastest is probably to fly to San Sebastian. The San Sebastian airport is right next to the French boarder, so there take a taxi to Hendaye in France and then train from there.

Standard route is 15+ hours. Train to Barcelona, then train to Narbonne, then train to Bordeaux, then train to Biarritz.

A bit more creative is train to Madrid Atocha, change to Madrid Chamartin, train to San Sebastian. That is 9 hours, then local transport across the border to Biarritz.

There are no direct flights from Sevilla to Donostia / San Sebastián, so you might want to fly to Bilbao Airport.
From Bilbao Airport, there are buses every 30 min to Donostia, where (after a short walk to the Amara train station) you can take the local train (Euskotren) to the French border town Hendaye and transfer there to a French train to Biarritz. It sounds complicated, but it should not take more than 3 hours to reach Biarritz from Bilbao that way.
Alternatively, a taxi from Donostia to Biarritz should cost around 130€ - significant, but potentially faster and more comfortable, considering that Biarritz train station is a ways out of town.

EDIT: I checked the wrong days, there are indeed Monday & Friday flights from Seville to Donostia airport. If it works for you, it is by far the best option.

Thankyou all it was very interesting and has given me food for thought