Take both, and some merino thermals. My daughter and I walked this route in winter, started in Ferrol in November and walked down to reach Lisbon a month later, in an attempt to stay ahead of the rain. It didn't work. On other trips to Portugal we have found it to be cold and wet from mid Sept onwards.
There are few refuges in the Lisbon to Porto section, but fire stations have stacks if mattresses and blankets, they tend to be used more by large groups going to Fatima, rather then solo walkers. But they will give you a basic place to stay. Cold though, as were most of the pensions and hotels we stayed in, for some reason heating is very poor in Portugal. My local friends laugh that its more likely your host will give you a coat to wear inside, then take yours to hang up. That said, lovely people and a lovely Caminho.