Last train Bayonne to SJPdP leaves 1806 29 March 2016.
SNCF TER Aquitaine TER 62 --- 13 Dec 2015 - 2 July 2016
http://cdn.ter.sncf.com/medias/PDF/aquitaine/62-AFFICHE NOV2016_LOGO_OLD_tcm50-35352_tcm50-9053.pdf
The last Chronoplus bus 14 you can take from Biarritz/Anglet Airport leaves 1721 and gets to Gare Bayonne 1800. This leaves 6 minutes to get from the bus stop to the station, buy a ticket, and board the train.
So try this: there is a train from Gare Biarritz leaving 1745 that arrives at Gare Bayonne at 1755. There is a Chronoplus bus C from the airport lv 1712/1727 to Gare Biarritz arr 1722/1737. (take a taxi if you don't make 1712)
SNCF TER Aquitaine TER 61--- 13 Dec 2015 - 2 July 2016
http://cdn.ter.sncf.com/medias/PDF/aquitaine/61 AFFICHE NOV2016_27_11_2015_tcm50-35349_tcm50-9052.pdf
Chronoplus
http://www.chronoplus.eu/
Your timings are tight. If your flight arrives late ...
-the train may leave late ... it waits for the train from Paris.
-the train leaves Cambo les Bains at 1834 ... so you might try taking a taxi from the airport to Cambo les Bains to intercept.
Use the SNCF station sites to get oriented before you travel.
SNCF Gare Biarritz
http://www.gares-sncf.com/fr/gare/frbiq/biarritz
SNCF Gare Bayonne
http://www.gares-sncf.com/fr/gare/frxby/bayonne