Sounds like a fun trip! Having a car should give you quite a bit more flexibility to stop as you go and hop out to explore for time to time, Northern Spain is great for that.
Regarding the Picos de Europa, we spent 3 nights at the Parador de Fuente Dé, back in mid-September of 2016, it was so convenient to stay right in the Picos de Europa National Park, the views were incredible. I really enjoyed the hiking in the area, just take the adjacent cable car up to the top, and there are a number of alpine trails.
It was light jacket weather when I was there. The restaurant at the Parador was fairly good as well, but the best dining experience was at nearby Meson del Oso, a rustic tavern/inn which serves the hearty local food of the mountain area. You don't need a room there to eat at their fantastic restaurant, but if they have rooms available this is also a good home base for the Picos de Europa.
We arrived late the first night driving direct from Santillana del Mar (also worth a stop). The next day we went up top of the mountains with the teleférico de Fuente Dé (an experience onto itself) and hiked all day. There is a restaurant at the top that you can have a quick lunch at. The next day we drove to nearby Potes to explore the medieval town and the immediate environs, we gave plenty of time for us to stop and explore an alpine stream or take photos of a beautiful meadow or two.
The last day we drove early to the stunning pilgrimage site of Covadonga aka the "Cradle of Spain", where the Reconquista was birthed after a fateful battle. The holy cave under the waterfall holds the tombs of the 8th century Visigothic Kings Pelagius and Alfonso I. There are good alpine hiking trails here too, up to some famous lakes in the high mountains.
This is just a slice of what one can do :)