Ooh, where is Grant when you need him? Grant, are you out there? Let's help Kate find out the train timetables to Bayonne.
Sorry Kate, when I went to SJPP last year it was a Friday. Yes, traveling on a Sunday can be challenging. What I know is there is a train that starts from Hendaye (it's on the border) that goes to Paris and the rest of France. This train stops at Bayonne where you can change trains to SJPP, but you can buy your ticket at Hendaye (last year it was 13.5 euros all the way). The train to SJPP leaves Bayonne around 1230-1 pm if I remember correctly.
Hendaye to SJPP around 3.5 hrs including the waiting in Bayonne. I suggest planning to arrive in Hendaye at around 9 AM.
Try a search for SNCF timetables. I hope this helps.
Mark