Yes, all the French places you mention can be very hot in summer, though certainly not at the same level as Andalucía. San Sebastian will likely be much, much more temperate. I did some checking before traveling there last June and found that on very rare occasions the temperature might spike, but it was typically for just a day or so, as opposed to potentially extended periods of 80-85-90+ degree weather elsewhere. The weather is not reliably warm and sunny on the Basque coast, so don't count on swimming every day.
From my perspective (not a beach person or bar-hopper) San Sebastian is not overwhelmingly blessed with things to do. I'd anticipate that you will be spending a lot of time making trips outside of town. Do you plan to have one or two cars, or will you be using buses and trains? Although I liked the Basque Country a lot, I'm a fan of scenery and pretty architecture. I'm not sure there will be enough liveliness and variety for a group including four non-adults.
Bilbao is worth a couple of visits for people who like art, or one for those who don't. The bus is much faster than the train, which is a narrow-gauge affair running through really nice countryside that I fear might be of little interest to the youngsters.
The interior capital of the Alava province, Vitoria-Gasteiz, has a large medieval district on a hill and several museums. Not very touristy. As of last summer one could take a tour of the cathedral, which was then (and probably still is) under renovation. That might appeal to the kids.
The border town of Hondarribia is picturesque, as are the towns of Zarautz/Getaria/Zumaia. I walked between a couple of those towns, right along the coast.
Farther away (overnighting best) you have the Altamira Cave near Santillana del Mar, which has prehistoric paintings. I don't know the details, but prior arrangements are needed for visiting it. Santillana del Mar itself is a very cute, historic (and touristy) town, but again I'm doubtful about how much interest it will hold for the typical teenager or sub-teen.
Beyond Santillana and inland is the Picos de Europa mountainous area, the most dramatic scenery I've seen in Spain. I've read that there are some active sports opportunities there. The old stone town of Potes is very pretty and may be the best base. Elsewhere in the area there's a mountain lift that gets you up into the peaks. This area is definitely worth investigating, but pay attention to the timing. Public buses and tour buses into much of that area do not begin until July. I don't know when the mountain lift opens for the summer, and white-water activities might operate within a very limited period.
Other affordable places where you don't generally have to worry about heatstroke are the Dolomites in Italy (where I doubt that you'd find a summertime swap), Normandy/Brittany in France (where your chances might be a bit better), or basically anywhere in the UK. London, of course, is a fabulous destination. If you could find a swap within a 1-hour commute... But you'd need to check out the price of transportation into the city to be sure getting six people into the city frequently would be affordable. There are two-for-one possibilities with the appropriate sort of rail ticket at some of London's sights, and others are free.
