I walked from SJPP March 10th this year, and I was worried about cold. Yes, sometimes it was...a beanie.buff/gloves are essential. And yes, the Valcarlos route was manditory...but it was also gorgeous (I was actually glad for the closure because I had been prejudiced and would not have otherwise chosen to go that way). And yes, until the at least the 15th, and usually the beginning of semana santa, lots of things were closed. The third day was rough from Zubiri to Pamplona because there was no-where to eat all along the way until getting into the outskirts of the city...and I wasn't prepared. So do look out for yourself that way. But by the time you're half-way, things will be opening and there's plenty of action.
You'll have to carry a heavier pack than the summer crowd (6Kgs?? How'd they do that??): I had 11, what with boots, warm clothes (more layers) and a heavier sleeping bag than needed when it's warmer. And rain gear is essential, but a poncho's enough. I had one that I got in SJPP and it was perfect.
That time of year is fantastic: there is just enough pilgrim company, and it's soooo lovely early in the spring, with the earth just waking up from it's winter sleep. All along the way, the apples and almonds are blooming, and when you get to Santiago, the Cherries will be too. Stunning.
(And maybe the bedbugs will not be hatching out yet?? Well, hopefully there are fewer of them...never saw evidence of a single one until nearly into SdC, and that was already mid April.)