So sorry to hear you're having these issues @Soubriquet. As others have said, a vet visit to check for underlying pain should be a priority - followed by a behaviourist that is (crucially) accredited - CCAB and APBC are two ones to look for
www.asab.org/ccab-register
www.apbc.org.uk/help/regions
Note that this is a behaviour issue not a training issue, so a trainer would not be appropriate. Some insurers will pay out for behavioural matters; I think PetPlan do for instance.
Depending on what the vet and behaviourist say, this may very well be a dog that does very well in a home that is - for instance - child free. Unless the dog has an underlying painful condition that's affecting welfare I don't think you should be considering PTS at this point in time.
However - when rehoming you would need to do so very, very carefully. My first port of call would be Dogs Trust - they have a selective intake policy so it's not guaranteed that they will take your dog, but once they take a dog, they will throw a great deal of resources at it from a behavioural and training point of view, and they really don't put a dog down unless it's seriously physically or mentally unwell - to the point that it's miserable and even skilled behaviour staff can't handle the dog safely using precautions. If they don't take a dog, at least you know where you stand before the dog is signed over. If Dogs Trust wouldn't take your dog, there are other rescues of course, but be aware that some will take dogs, assess them, and depending on the outcome of that assessment will PTS (e.g. Battersea, RSPCA). Don't dismiss rehoming kennels until you've been to visit - some of them are really very nice.
I certainly wouldn't be rehoming privately unless you personally know a behaviourist who wants a new pet dog (and challenge) or children are identified as a specific trigger and you have friends / relatives who are experienced dog owners and childfree, and who agree to return the dog to you if it doesn't work out - you'd be passing the buck onto someone else, and that tends to be the way that dogs get passed around, as there's no back up support.