Basque country

The temperature in the Basque Country even in mid-summer tends to be very moderate, so it's a great destination for folks who need to travel in July and August. Even then, it may be quite overcast or rainy. You can explore actual day-by-day temperatures, usually going back more than 10 years, on wunderground.com. Choose History > Monthly. I have never researched shoulder-season weather in that area. It's possible that it's not terribly different, but I suspect there is substantial rain. The Basque Country is lovely and green.

If you don't stray too far from the coast, most places between the Basque Country and Galicia (A Coruna being an exception) tend to have similar weather, so they would fit well with the Basque Country on a summer trip. If you get as far south as Leon (otherwise a very attractive destination), I think serious heat is a risk.

Northern Portugal was not hot when I traveled there in June, but that was many years ago, before the impact of climate change was so obvious. I have no information about current conditions there (but wunderground does).

Considering that I spent time in Bilbao as a day trip from San Sebastian I would say the summer months are great as it does not get hot or humid and there are a lot of out door festivals in the evenings while you sight see in the day and the days are long during the summer months

Spain and Portugal have sparse connections, so before planning on getting from Bilbao to Lisbon to connect these tours, investigate the flights (and unless you really fancy many hours on multiple buses or trains, you will have to fly).

I was in Basque country (on my own, not on a tour) in mid-May 2017. The weather was very nice, but it was made clear to me that this was not something to be counted on. The area is famous for its rain (like Ireland, there's a reason the landscape is so green), and much of the year is cool and rainy. Summer, on the other hand, doesn't get nearly as hot as the rest of Spain - that's why even Franco summered in San Sebastian.

It was also very clear that I was there before "high season," which starts June 1. Things were still running on "shoulder season" hours, like bus tours or store opening hours and days (more things were closed on Sundays than would have been the case in high season).

We spent a week in the French Basque country 5 years ago, starting in mid-March. From what I remember, the temperature was very pleasant, but rain showers often kept us from sitting outside on our terrace. And from what I read in other comments here, rain is common there. The countryside was beautifully green, so that explains it.

San Sebastian is a must see. I liked it way better then Bilbao. Crowded in summer but the shoulder seasons May Sept-Oct should be ok.

I preferred Bilbao with its larger historic district and the tourists mostly confined to the Guggenheim. In San Sebastian, the smaller historic area seemed to be full of overrun tapas bars. Fortunately, the two cities are close enough that travelers can visit both and come to their own conclusions. And the inland capital of Vitoria-Gasteiz is also very nice.