I've been watching with interest the river levels here: https://www.vigicrues.gouv.fr/niv3-station.php?CdEntVigiCru=32&CdStationHydro=Q935251001&GrdSerie=H&ZoomInitial=3
And webcams of the rivers here: https://www.bayonne.fr/ma-mairie/webcams
Express Bourricot finish at 18:00 so you could book a private transfer with Biarritz Airport Transfers. If your flight lands on time you can be leaving the airport within 15 minutes (don't know if you are flying national or international). Then the journey is 55 minutes to 1 hour.
You can get the TxikTxak bus number 3 or 4 directly from the airport bus station. After you leave the luggage collection area, turn left and walk to the far end of the airport building. Leave via the sliding doors and there is a covered walkway to the bus station.
Something fell on one of the tracks yesterday so it caused huge delays. News from the local paper (only in French): https://www.sudouest.fr/economie/transports/circulation-en-gironde/sncf-la-circulation-des-trains-tres-fortement-perturbee-ce-samedi-dans-le-sud-ouest-16312906.php
I would get the SNCF train from Biarritz to Hendaye then, depending on where you are staying in Irun, you can walk across the border to Irun or get the Euskotren to whichever stop is closest to your accommodation.
I've been there a few times and the welcome is always very friendly. Be aware that it is not in Saint Jean Pied de Port but in Ispoure, about 2 km from the centre. The address is actually quite hard to find on Google Maps: 123 Rte Itzalgui, 64220 Ispoure
Bus from Biarritz Airport to San Sebastián with ALSA or Flixbus (be aware the Flixbus stop is no longer at the airport).
Bus from San Sebastian to Pamplona with ALSA.
Bus from Pamplona to Puente La Reina with La Estellesa.
The quickest way would be to go direct and reserve a private transfer...
San Sebastian airport is very small so assuming you have no luggage to collect you can be through arrivals 10 minutes after the plane wheels touch the runway. At that time on a Saturday the journey from the airport to Bayonne will probably take 40 minutes so you can probably just make the 14:20...