Ummm... (hoping to not fan any flames...)
I have had several dogs, the first was just a mutt... small brown dog, part dachshund (sp?), part... well, heinz 57. He was the cutest puppy. Found him along the road, where someone had dumped him. I named him Weasel. We shared space for many years. (One never 'owns' a pet.)
Weasel grew up outside. He was an outside doggie. If he were co-oerced into the house, he got 'nervous', and piddled. If I picked him up to carry him, he got nervous and piddled. I accepted that as one of his characteristics. I never tried to 'desensitize' him. He didn't need it. *grin*
Weasel loved the outdoors. He knew that I lived inside, and he lived outside. We both understood and accepted that arrangement.
When my daughter was growing up, we had several dogs come into our lives. Her first was Alice. Alice was a 'give-away' dog. The ex-wifes' sister had this dog, and the husband was always spraying the dog with water, making the dog cower, etc. They were a 'throw-away' mentality type of family. If they got tired of something, living or not, they threw it out.
We were very fortunate to have been given the opportunity to get this dog, but I stipulated from the start that ANY dog had to get along with the cats that we already had around the house.
Everyone understood - and fortunately, Alice only chased the cats for a short time, and then settled in. We pretty much left the dog (or any animal) alone, and their own little personalities seemed to flourish this way. It turned out that Alice was not such a 'bad' dog as we were led to believe. She and my daughter (about 4 at the time) bonded really well, and we had that dog all the rest of her life, when she died at an old age of 13 or so. (Often times, the former owner would come over and lament having given the dog away, but I knew that the dog would not have blossomed like she did with us if she were still with the other person.)
One dog that we had, from a puppy, was a very large dog ('horse' would be a better description). Tiny was her name. She was a machine of destruction. She would chase the cats... who probably needed the exercise, and she would also chew up everything in the yard. That was not something that I liked to see. We had to keep this dog on a leash, attached to a post anchored into the ground (she often was strong enough to bend or break this anchor, too). We gave her 'yard time' every day, and the other dogs would 'groan' whenever we let her off her leash to run 'free' for a bit.
When the decision was finally made to give her away (an ad in the paper), the daughter was in on the decision. She understood the reasons why we had to give the dog away.
I would recommend that your son be allowed to be with you when you give away the doggie. It will be a tearful moment, but he will get to see where the dog is going. It _can_ be made into a 'positive' event, perhaps by explaining that it is better for both the doggie and you, that way. It will also be a 'real' thing, him getting to see the new owners, and the doggies' new home.
Just my .02 worth.
Regards,
Jim TX