Rome2rio is a great source of information, but it often misses local bus lines, including what may be your most efficient option. Carexpress bus company's #3 bus goes directly from Biarritz airport to the Hendaye train station ("Gare"), from where you walk over the bridge to Irun, as noted above.
Their schedule shows buses every 2 hours this Sunday (more on other days of the week). Length of travel is 1 hour. This will probably be a lot more efficient than taking a bus into Biarritz, Bayonne, or St. Luz, and waiting around for a train connection to Hendaye.
Here's the web site for Carexpress:
https://www.carexpress-hendaye-bayonne.fr/#nav
Btw, I found this option using Google maps, which allows you to search for public transit options from place to place. I find that for local travel, either within a town or between nearby towns, it often gives you more precise options than does Rome2rio. In any event, try both.
For anyone else reading this post, there seem to be other direct bus options with Alsa bus and Pesa bus, which also go on into SpaIn, but not on Sundays, which is when the OP is traveling.