We left SJPdP on March 15th this year. We had no trouble finding albergues to spend the night. Also, I found Gronze to quite reliable about seasonal start dates and daily opening times.
In terms of the weather, you may want to look at the weather forecast a few days in advance of your departure and plan where to stay, especially the first night or two. We got lucky with the weather this year and made it to Roncesvalles the first day. If we had left two weeks earlier, we would have run into snow that would have made staying in Valcarlos absolutely necessary.
Lastly, we had booked a flight into Madrid early on. Our plan originally was to take a train to Pamplona and then the bus to SJPdP. When I looked into tickets for the bus, I found out it didn't run that early in March. We could have taken a taxi but we opted for a more circuitous option -- fly from Madrid to San Sebastian, walk from the airport through Irun to Hendaye, spend the night in Hendaye, train the next day to Bayone and then Bayone to SJPdP. It worked out fine for us, but flying into Paris would probably be simpler for you.