Burgos to biarritz

The border crossing means no direct services from Burgos to Bayonne.

Note on trains: Irun/Hendaye are break of gauge stations. France and Spain operate trains on different gauges and at different voltages. A transfer from Renfe to SNCF trains is required.

Easiest option is Renfe train Burgos to Irun/Hendaye. SNCF train from Irun/Hendaye to Bayonne.
Some Renfe trains terminate Hendaye others at Irun. SNCF TGV and Intercites trains leave from Irun. TER trains leave from Hendaye.

San Sebastian is the closest city to the border.

ALSA operates buses from Burgos to San Sebastian. They provide a limited service from San Sebastian to Irun Renfe station and a transfer is usually needed.

There are many transit options from San Sebastian to Irun and Hendaye. The Euskotren leaves from Amara station to Hendaye. The Ekialdebus leaves from a bus stop near the San Sebastian bus station and has stops near Irun Renfe Station. Renfe Cercanius runs from San Sebastian Renfe station to Irun Renfe Station.

ATCRB buses operate from Hendaye to Bayonne Place de Basques.

pre booking is going to create additional pressure on you. Just wait until you get to Burgos or whatever your ultimate destination and go from there