Hi, Andrew,
I agree with a lot of what Dave said. One thing to keep in mind is that the numbers posted for the Camino Primitivo are unreliable, at least according to what I've been told. That's because all of the people starting anywhere between Irun and Villaviciosa are counted as being on the Norte (Villaviciosa is the turn off on the Norte for the Primitivo), even though many of them walk from Irun to Oviedo to Santiago.
I walked the Primitivo last June and the traffic was perfect for the available albergue space. I met some Spaniards who had started in August of the previous year and had stopped after a couple of days because of huge crowds. Unlike the Frances or many parts of the Norte, there's not a solid backdrop of private accommodation for use when the albergues get full. So as Dave says, that will mean overflow space on the grass or in sports complexes.
If you can go earlier in the summer, that would be my definite preference. I've walked it in both June and October and both were great times of year. Buen camino, Laurie