I really feel for you op. And I'm disgusted by the "rescue" putting your family at risk like this. I'm in a similar position in that my dog has a lot of issues and is completely reliant on me. I have some experience but was unprepared for the impact on my life. It can feel suffocating at times and very isolating. And I don't have kids, or health issues on top.
I'm no rescue expert so can't really suggest much but I think there is a place in the NW that takes unplaceable dogs; they don't use kennels and I think they offer a home for life. I'll try and find the name.
I appreciate you may be past the point of no return but did you see a veterinary behaviourist? That has made all the difference for me. I saw other (accredited) behaviourists before and they were pretty useless tbh. It's surprising you're not still working with the behaviourist, as the dog is clearly not better! The vet behaviourist has prescribed medication for my dog and a behaviour modification plan; we are still working with them 12months in. They are also checking for physical causes, e.g
pain, gastro issues etc. Is it worth asking your vet for a referral? I have claimed all treatment through my insurance. But it is a lot of work and I'm not sure I could do it if I had a family, or would even want to.
In your position, the immediacy of my decision would depend on if I felt my family were at risk. If that is the case, the rescue won't help and you can't find a suitable alternative, perhaps you need to have a chat with your vet about euthanasia. The poor dog must be miserable living in so much fear and there are worse things than a quick peaceful end. If the dog is a bite risk, I would question whether it's even responsible to try and rehome it, unless somewhere very specialist that can guarantee they will not pass the dog on. Even then, you have to wonder what the dog's quality of life will be if its so distressed. For me, I feel that by going through the vets, I am exhausting all options but, again, I'm not worrying about my kids being bitten. I hope you find a solution for your family and I'm sorry you've had such a horrible experience. We live and learn but it's fecking hard sometimes! x