If you are hoping to encounter fewer crowds, I think the data shows September from SJPP might be less crowded, particularly as you are planning on walking all the way to Santiago. As you know, the closer to Santiago you get the more people you are likely to encounter. This will be particularly true in mid-late June.
On the other hand, in Spring things are greening up and just generally look better. I think it's likely to be hot for part of your journey at either time.
From my experience, the crowds tend to bunch up going into and out of some of the larger cities like Pamplona, Burgos, Leon and even Sarria, They spread out again not long after. A way to avoid the some of the crowding is to stay in "non-
Brierley" villages, that is the villages on either side of sections recommended by Mr.
Brierley. No matter when you end up going, hey, you are going! That's pretty great.
Buen Camino