I think of two pilgrim friends in particular who have done the Camino Francés more than once with tents (I've met many other people along the way with a tent, but these ones stand out most to me). One is from Hungary and camped in September/October the first time he went (frosty mornings by the time we got to Galicia!), and the other is from Northern Ireland. He's walked it in autumn and in freezing cold spring, and last year he and his girlfriend cycled the Camino with a tent (and they leave home next week to do the same again!)
They're respectful and careful each time they wild camp, and they find a different place if they're asked to move on.
Yes, their gear weighed a lot, more than I've ever thought I could hope to, or would want to carry. But they've managed, and maybe that has a lot to do with the fact that they wanted to make it work. They're two of the most resourceful, determined people I've ever met.
It's certainly not necessary to take a tent. I've personally, in four Caminos, only arrived at a full albergue once, and that was at 7:30pm (Ferreiros, a small municipal albergue in Galicia; my friend and I walked on to Portomarin and that was another great Camino adventure
). But if you want to, it's doable. Maybe you can leave home something else from your pack to balance the extra weight gain?
Buen Camino, whatever you decide! It's freeing, I think, to know there's no wrong answer.
Rachel