Hi, Beavers, and welcome to the Forum
After my arrival in Madrid this September, I decided that I will be taking a short flight from Madrid to Biarritz airport. From Biarritz it's just a 40 minute shuttle ride to St Jean Pied de Port. When I looked at taking the train to Pamplona and then getting transported from Pamplona to St Jean Pied de Port, it was a longer trip and didn't save me much money.
When looking at transport options from Pamplona to SJPdP, I found
this taxi company in Pamplona which specializes in taking pilgrims to SJPdP. You make an online reservation for pickup at your arrival location. During the reservation process, you have the option to select that you are willing to share a taxi with others going to SJPdP, so that the fare is split up. The company will work to schedule a full taxi for you, or assign you to a taxi from someone else's reservation to share.