There is an excellent bus service to Logrono from Bilbao (Termibus), several departures every day. Also, there is a bus from Biarritz/Bayona to Bilbao.
I have no actual details though, you must google a bit.
I am 51% convinced it would be the quickest way to use the bus connections, and not by train through Irun.
BTW, I would like to mention the excellent public bus services all over the Camino area. -Which is reassuring when you are afraid you might get into trouble while hiking, healthwise or other. There is always help and transport nearby.