Justine, I did not take camping gear walking from Le Puy to Pamplona, but wished I had. I did take it for my second Camino, Pamplona to Santiago and beyond, but the Frances was not really suitable for camping out.
For example, if I'd had camping gear on the Chemin du Puy, I would have been able to do the Cele variant (accommodation was all booked out by a large French holidaying group.) Also, I could have stayed in camping areas adjacent to gites, which would have enabled me to stay near friends but away from their snoring (snoring, alas, being a very big issue with me).
I think the fact that the Le Puy route doubles as the GR65 makes it a touch unpredictable for accommodation, and camping gear - in my case, just a sil nylon tarp and short concertina mat - can give options. Not that one should worry too much about accommodation on the Chemin du Puy.
For a sleeping bag, I'd favour a light down mummy, since I experienced freezing conditions in May in the Bearn, though I expected heat to be the biggest prob after the Aubrac. A bag like that is versatile: unzipped, it can be used as bedding indoors. I dare say there are new products that will serve as well or better, but I would avoid bulk without underestimating the chances of cold weather.
Hey, the main thing is you are going to do both Caminos in one go. My first Camino friends out of le Puy were quebecois who did the whole thing without a break. (Like most members of the human species, they moved a lot faster than me.)
What an experience! I am truly excited for you.
Rob