Bibao to St Jean Pied de Port with Bicycle

You might want to consider using Tour 'n Ride who operate from Santiago. They deliver the bike to you and then you ride it back to Santiago.
Or:
The bus from Bilbao to Bayonne goes twice a day (at 7.30 and at 1300 hours). The company is PESA http://www.pesa.net.
ALSA buses to IRUN leave Bilbao many times a day. Then you can then travel from IRUN to BAYONNE by bus/train and from BAYONNE to SJPP by train or bus. http://www.pesa.es Coach service to Hendaye.

BUS: ALSA is the main bus service in Spain. When booking your ticket on ALSA you must indicate that you will be travelling with your bike. The system will automatically add the bike fee of €10 (long distance) to your ticket. Passengers taking a bicycle must be present for boarding with their ticket at least 15 minutes prior to departure.
Because of limited space available they will only carry 4 bikes per bus. They must be packed into a box or bag. If there is no more space for bicycles in the bus, the box offering a bike selection will not be available.

Train: It is permissible to carry bicycles on all middle distance trains in Spain with a specific area for transport.
Media R-598 Distance (MD) trains allow three bikes per train. It is necessary to obtain a ticket for the carriage (free), plus the passenger ticket.
Regional and Regional Express trains allow one bicycle per passenger if space allows and the maximum allowed for each car or train is not exceeded.
TRD trains do not allow the transport of bicycles unless they are folded with maximum dimensions of 30 inches wide, 66 inches high and 82 inches long when folded.