Hi,
A shared taxi is the fastest (and it can be cheapest) way to get from Biarritz airport to SJPDP. I travelled to SJPDP last year - flew into Biarritz, got the bus to town and then the train to SJPDP. A few people approached me at the airport to ask if I'd like to share a taxi to SJPDP and I said no as I'd booked the bus and train tickets already. I later regretted it as it would have been so much faster, and the same price, to share the taxi instead of get the bus/train. On the plus side - it's a beautiful train journey. So it's a win win. And SJPDP is beautiful too.
My vote - shared taxi.