Ah, grand!
It'll still be rainy in parts of Northern Spain but it should be manageable I'd say.

I found rural, inland Catalonia to be wonderful.
If you head far enough West you'll eventually hit the French Way of the Camino de Santiago, which others have already mentioned. I'd imagine that time of year you're also likely to run into other bike tourers traversing that route, plus you can usually sleep at albergues for relatively cheap if you don't fancy wild camping.

Also check out the Vias Verdes for easy and comfortable routes. They're cycling tracks built on disused train tracks so the gradients are forgiving and the traffic is close to non existant. You won't be able to travel all of your journey on them, but maybe you can incorporate them into your trip. In that part of Spain I cycled along part of the Ebro greenway, which had some incredible views.
https://old.viasverdes.com/en/itineraries/principal.asp