A few months ago I had to research some travel options for a dog-owning couple who were visting the UK. Nearly all of the ferry companies require dogs to be housed in kennels (or in their owners' cars) on a nasty, smelly, scarey car deck. However I was struck by the dog-friendliness of Brittany's Portsmouth to Le Havre service, where they've got cabins which you can take your dogs in, together with a dedicated pet exercise area. So, given that you're starting from Hampshire, that would seem to be a possibility for you to consider:
http://www.brittany-ferries.co.uk/ferry-routes/ferries-france/portsmouth-le-havre
That would obviously give you a longer drive than from Bilbao or Santander but the crossing would be far cheaper. You could either take the A10/A-23 route, down the western coast of France and across Spain, or the A75/AP-7 route, cutting across France and then along the southern coast of France and the eastern coast of Spain. You'd have about 16 or 17 hours of driving on the Continent, so that's either a couple of very long days or, probably better, three fairly relaxed days (with time to stop and enjoy some sights).
https://www.viamichelin.co.uk/
(For comparison, it's about 9 hours drive from Santander to Alicante)