I walked for a while along the del Norte with a Slovenian pilgrim and her companion and their dog. They were allowed by most albergues to camp in their yard. There are few places for wild camping along the del Norte-- almost all land is agricultural or urban, and Spanish law does not permit wild camping within (I think- someone can confirm this) 3 kilometres of an existing campsite. I note that you state that you love the wild, but the Camino del Norte is not a wilderness hike, but an extended walk through a mix of forests, hills, farms, villages, cities, and beach-side resorts. As much of the spiritual benefit comes from comradeship with other pilgrims and encounters with the Spanish as it does from the (impressive and wonderful) landscape-- but as most of the spiritual experience is very subjective and individual, most comment will naturally focus on the practical aspects, as we all share in that. I wonder if other Camino routes, such as the Primitivo or the Vadiniense, would not suit a wilderness hiker better.
There are several threads on this site about the ups and downs of taking a dog on the Camino, and opinions are very strong on this, but if you read through these threads, you will find that there has much thought been given to this. I tend to fall on the negative side as, of the dogs I have met on the Camino, possibly two were having an enjoyable time, and others were definitely not having a good time. They were suffering, sometimes considerably (bleeding paws, exhaustion), but doing their best to loyally keep up with their masters. Most guard dogs are chained or behind fences, so your dog is less likely be be injured or killed than many think, but there remains a problem with shepherds' dogs, whose first instinct is to protect their sheep.
You are to be commended for rescuing the puppy from a difficult situation, but I am not sure if an extended walking trip is the best idea for the dog. I may be wrong. Perhaps, as suggested, a home for the dog while you walk is the way forward.