So many posters here need a serious reality check, and have not properly read the OPs posts.
This dog was very diffcult and reactive before the kids came along. This is not a case of a nice dog reacting to having his fur pulled. This is a dog living in constant anxiety, having to be kept behind a gate/door from the family, unable to have visitors to the house, needing to be muzzled, scared and reactive outside, cannot be walked off lead etc etc.
There has been 3 years of behaviourist input and the behaviourist has said that he will not get any less anxious. OP is an experienced dog owner, and has done all the right things and STILL 5 years later, this dog is anxious and difficult.
If this was just a case of dog not mixing with kids, that would be different.
The dog is from a rescue. No other rescue will take him, they will insist that he goes back to the original rescue, because that will be in the contract when the OP adopted him, he remains the property of the original rescue.
The original rescue has shown itself to be unhelpful and frankly crap. They would not help or support.
OP has clearly said that on their website they do not give any negatives about the dogs, only the positives. I would not trust that rescue to rehome to a safe home. Which means that he will be rehomed (eventually) to a home where he will probably bite and hurt someone. Or be pushed from one home to another, which is really really unkind to this anxious dog.
There simply are not homes around for all these highly reactive difficult dogs.
One PP said they had a young cocker spaniel who had bitten, but was then able to go on to a long happy life. THAT IS 100% DIFFERENT. I have a rescue dog who in the wrong context would bite. But he is NOT like this dog, reactive, scared anxious and very, very difficult.
Those saying rehome him to someone who can work with him, he has had 5 years of work. There are just not enough homes around for these dogs, and he will stay in kennels for months waiting to find that mythical home.
In that time he will be unhappy and stressed, and get worse.
I simply do not understand those who say that no dog should be PTS. I actually would now never use a rescue that had a no PTS policy. We, as responsible owners, should all recognise that for some dogs, being PTS is the kindest option. Why is living a life scared and stressed better than a quiet death?
Dogs can be mentally ill as well as phyisically ill, and putting them out of their misery is sometimes the kindest thing.
This emotional cry of - don't put to sleep - is the worst kind of animal care, based on our emotions and not on what is best for the animal. Most vets would not hesitate to PTS a dog who has bitten.