Hopeless??

[Gwinnywillow]

There is a shot the vets can give that keeps a dog from having skin irritation for about 3 months. Perhaps you might look into that?

[lpollockandrsn]

It sounds like you are doing a lot to help this little guy. If the vet thinks there may be something more neurological you might try a behavioural medication. There are some that seem to help with anxiety/behavioral problems, like treating a Tourette's case in people.
I totally agree with trying to change the diet to find the right balance but as you are finding that he is not suffering from allergies it is a long shot. Of the dogs that are 'allergic' to something, less than 10% are allergic to what they eat, according to the allergy laboratories like Heska. We tend to humanize that aspect a bit since so many people have food allergies. Not as true with dogs.
I think fatty acids and supplements help if the coat and skin are dry.

I am sorry to hear that euthanasia might be an option. I would look into the medication first if nothing else is helping. Most are not very expensive, and I have known them to help a lot for a variety of problems including self mutilating behaviour.
I have two young boys too and I can only imagine having that conversation with them. How old is your son? Let him know you are doing everything you can to help heal the problem and it would only be a last resort if his body continues to give up on him.