Unfortunately we were in a situation like this several years ago. My husband likes to hunt occasionally and wanted a bird dog badly. On an impulse, he went out and bought a German Short Hair Pointer. This dog was a HORRIBLE fit for our family. She was destructive, chewed everything in sight (well beyond puppyhood), barked incessantly and was the most hyper dog I've ever seen. It wasn't her fault, it was DH's for not researching properly to make sure it was the right breed for us.
We did everything we could to make it work, including consulting trainers and behaviorists, and DH would go out every night after work to just let her run in the field and burn off some energy. She was just the type of dog that was bred to have a job (hunting) she wasn't happy unless she was doing it.
We eventually came to the conclusion that it wasn't working for us, and no one, including the dog, was happy. It was time to find her a new home. I'm involved in Great Dane Rescue myself, so believe me I felt guilty about this decision. Under no circumstances would we have turned her over to a shelter or rescue, we just knew we had to find a home for her that would be a better fit. We went back to her breeder for help and a few months a man called us who was the perfect match for Oreo. He was an avid hunter who worked outside and could take her with him all day. Long story short, we still keep in contact with him and it's been a happy ending for everyone. Our house is a lot calmer, the guy has a great new companion, and Oreo has a wonderful new home where she can do what she was bred to do.
I'm not proud of the fact that DH bought a dog without thinking it through more clearly, or that we had to eventually admit defeat, but I am happy that we were able to find a better situation for her. I really don't consider it abandoning her as we waited months until we found the perfect new home and would have kept her forever if necessary. Sometimes things just don't work out, and as long as an owner is responsible in finding an appropriate new home, rehoming may be the best solution for everyone (including the dog.)