Only God and the weather determine when the pass is actually open. April 1 is a relative date for reference. You ought not rely on that for making hard and fast plans.
My suggestion, if you are fixed on starting at SJPdP, is to make three alternative plans:
Plan A - the pilgrim office tells you, when you arrive, that the Napoleon Pass is open or will be tomorrow... follow Plan A
Plan B - plan to follow the Valcarlos Route from SJPdP to Roncesvalles, many have done so. The difference is that this route more or less follows the serpentine road over the Pyrenees from France into Spain. So, it is kept clear of snow all year. There is some road walking. Research the Valcarlos Route here in the Forum.
Plan C - pre-investigate and make reservations as needed to have Express Bourricot drive you over the mountains to Roncesvalles the next morning. Plan to be taxied over in the morning, after breakfast at SJPdP. Then you can start fresh walking from Roncesvalles to Zubiri, or onwards towards Pamplona. The drive over the mountains is about 45 minutes, and is not for the faint-hearted. I have driven it in summer weather. It is very windy and equal to the world's best amusement park rides.
Failing all of this, and if you are open to it, consider starting at Pamplona. Personally, I plan to fly there from Madrid. Coming out of Florida, I fly from Miami to Madrid on Iberia, then connect in the same Terminal 4 to Pamplona. There are several flights daily. For me, flying into Madrid is straightforward and easy. From there I fly to Bilbao, Barcelona, Santiago, etc. with ease.
Hope this helps.