recommended end point one week trip

Walk the Camino Baztan http://caminodesantiago.consumer.es/llevatela-al-camino/completa/?camino=camino-baztanes from Bayonne to Pamplona. Take the bus from Pamplona back to Biarritz http://www.rome2rio.com/s/Pamplona/Biarritz

Eight days will get you from SJPdP to vicinity of Logroño.

Take Renfe train from Logroño to Bilbao Abando Indalecio Prieto; Or,

Take La Union Y Burundesa bus from Logroño to Bilbao Termibus.

Bilbao Airport has connections to Dublin.

Getting around in Bilbao:

Take Metro from Bilbao Abando Indalecio Prieto Renfe station to Bilbao San Mames. San Mames is co-located with Bilbao Termibus

Take Local bus from Bilbao Termibus to Bilbao Airport BIO

Renfe
http://www.renfe.com/EN/viajeros/

La Union y Burundesa
http://www.autobuseslaunion.com/en/

Bilbao Abando Indalecio Prieto
http://www.adif.es/es_ES/infraestructuras/estaciones/13200/informacion_000034.shtml

Bilbao Termibus
http://www.termibus.es/index.php?idi=en

Bilbao Airport
http://www.aena.es/csee/Satellite/Aeropuerto-Bilbao/en/Bilbao.html

You can walk all the way from Biarritz to Bilbao in 8 days and take a non-stop flight back to Ireland from Bilbao (they aren't daily though so check schedules).

Vía de Bayona from Biarritz to Hendaye (you can look for info about it on the forum using the search feature on the upper right corner). Once in Irún (next to Hendaye, on the other side of the river), continue on the Camino del Norte till Bilbao. You'll find plenty of info about the Camino del Norte both in the Camino del Norte área of this forum as well as in other webs.

An alternatively to the Vía de Bayona, would be to walk the Sentier du littoral from Bidart (next to Biarritz) to Hendaye (next to Irún). For info about the Sentier du littoral: www.guethary-tourisme.com/discover-guethary/path-of-the-coast.php