Hi
I have done this already (Flew to Bilbao, then bus and train to Bayonne, SJPP, later back from Burgos to Bilbao)
If you haven't been there yet: Biarritz and Bayonne are practically one city.
IMO best connection:
1. Bus from Burgos to Bilbao (roughly every hour, ALSA, 1:50 travel time...