Last ATCRB bus leaves Gare d'Hendaye 2022 Saturday arriving Place de Basques Bayonne 2135.
ATCRB
http://www.transports-atcrb.com/ligne-bayonne-hendaye/
Bilbao to Irun
The last ALSA bus that arrives at Irun before 2022 leaves Bilbao 1830 arrives 2010. That leaves 12 minutes to get from Irun Renfe Station to Gare d'Hendaye ... 3km. You'd have to have a taxi to do it.
The previous ALSA bus leaves Bilbao bus station at 1720 and arrives 1905.
Euskotren leaves from Irun 1918 and 1948 arrives Hendaia 1922 and 1952. Euskotren Colon station - leave Exit doors from Renfe Station (where ALSA bus terminates), turn right (North) cross bridge across the tracks and find entrance to Colon Station on Left (North) side of street.
ALSA
http://www.alsa.es/en/
Euskotren
http://www.euskotren.es/
Bilbao to Hendaye
The 1700 Euskotren leaves from Axturi station and arrives Hendaia at 2022. I would hope the ATCRB bus waits ...
However, getting from the airport to Axturi station would likely require a taxi.
The previous train leaves at 1600 ... no chance.
Airport to Termibus
The airport bus goes to Bilbao Termibus leaving every 20 minutes. The last bus you can catch is the 1640 bus which arrives at Termibus 1710. See Termibus for Schedule. You'll have 10 minutes to catch the ALSA bus.
Termibus
http://www.termibus.es/
Bilbao airport
http://www.aena-aeropuertos.es/csee/Satellite/Aeropuerto-Bilbao/en/Home.html
Airport to San Sebastian/Donostia
PESA operates buses hourly from Bilbao (Loiu) Airport to San Sebastian. The 1645 bus arrives at San Sebastian bus station at 1805 (Pio XII). The 1745 bus arrives 1905
http://www.pesa.net/pesa/public_home/ctrl_home.php
San Sebastian to Hendaye
From San Sebastian Bus Station go 700 meters to Amara Euskotren Station. Catch 1845 Euskotren to Hendaye arriving 1922 or 1915 arriving 1952.
Renfe operates Cercanius from Renfe Station San Sebastian. 1925 - 1948 arrival at Irun Renfe Station. That allows half an hour to catch a taxi to Gare d'Hendaye. Cercanius is faster than Euskotren but only terminates in Irun.
http://www.renfe.com/viajeros/cercanias/sansebastian/index.html
Conclusion
The best option is to go to Bayonne Saturday and catch the 0745 SNCF bus from Gare de Bayonne to SJPdP Sunday.
The last ATCRB bus leaves Gare d'Hendaye at 2022 and arrives at Place de Basques 2135.
The last suitable Euskotren leaves San Sebastian/Donostia Amara station at 1915 and arrives at Hendaia (colocated with Gare d'Hendaye) at 1952.
The last suitable PESA bus leaves the Bilbao (Loiu) Airport at 1645 and arrives at San Sebastian Bus Station at 1805. You might try the 1745 bus arriving at 1905 but that leaves 10 minutes to get to Amara Station to catch the Euskotren or 20 minutes to catch the Renfe Cercanius to Irun
1530 arrival should leave enough time to get the 1645 PESA bus to San Sebastian. Timings are a bit tight.