This is a difficult one.
Public Transport links between France, the north of Spain and the Algarve are not good.
From Biarritz, apart from car, you probably are restricted to train or long distance coach.
Train:
1) The TGV/Sud Express leaves Biarritz at 20:49, arriving Irun at 21:23.
Departs Irun at 22:00 and arrives Lisbon Oriente at 10:54 the next day.
2) There is a connecting train to the Algarve departing Lisbon Oriente at 13:20, arriving Faro at 17:22 or Lagos at 17:55.
Buses:
Eurolines Portugal have a service advertised from Bayonne to Faro here:
http://www.eurolinesportugal.com/horarios_tarifas/ht_08.html
Although the service shows a timetable for winter 2007 I understand that the service still operates.
Dept: Bayonne at 23:55
Arr: Faro at 18:15 next day.
