Pau to Jaca

I drove over the top from Lourdes last week on the next pass to the east from the one you are talking about (Frontera del Portalet). There was only a little snow at road level but all the ski resorts were busy with people taking lifts up to the snow. I was in a 2x4 pickup. No opportunity to test a new set of chains bought in Lidl (UK) for £20 last month. lol. I'd think the N137/E4 would be fine, even after heavy snow. They are organised with snow ploughs.

I drive that way every year in April. Usually there are no problems with snow. The Somport Tunnel is wonderful and cuts off the mountain summits where there is a risk of snow.

The Aspe Valley on the French side is a bit narrow in places but usually no difficulty. It's a great route to avoid the traffic and tolls on the Basque route.