There are so many responses already, I admit that I did not read through them all, so you probably have already heard anything I have to share.
I can understand your unwillingness to keep the dog in your home because of what has happened, but also APPLAUD you for not taking the easy way out and having the dog euthanized.
I have worked in animal rescue for many years, and have lived with as many as 15 dogs and 10 cats at one time.
Something to consider is that the animal was in pain. While you say that the tail being clipped in the door has happened before - each incident is individual. The door may have been shut with more force or on a different part of the tail than other times. There could be other factors as well - had the animal just eaten? Maybe he was tired and thus a bit cranky. Perhaps there is an underlying illness that you are unaware of, or maybe his tail had already been shut in the door that day and the second time was more painful. the truth is, you weren't there, and although your 19 yr old neice is probably very responsible - she was there caring for multiple children - as all of us mothers know - you can't always see everything, everywhere - so one of the children may have injured or upset the dog at your neice may have been unaware of it.
Instead of putting the dog in the bathroom, another solution would have been to purchase a dog crate and use it (anyone else reading this who has dogs SHOULD crate train their animals - do a search for crate training to learn more about it, it will take too much space to explain here.) Dogs that are crate trained learn that the crate is their safe space. The dogs that I have shared a home with (various breeds, ages, sizes) have always been crate trained. All of that being said, I do understand not wanting to keep your dog - you will always be afraid of another attack and animals can detect fear. before the dog is rehomed, he really should be taken to the vet to determine if there was anything else that could have contributed - like an illness or injury. if the dog has never bitten before and has always had a good temperment, chances are there is something else that you are unaware of. This dog should be given every opportunity to find a new home and should not be killed because of one mistake (I understand there were multiple bites, but it was one incident).
I have been bitten by dogs in the past (when you're active in rescue, things happen) - and ALWAYS - the fault was NOT the dogs.
One dog had a sore on his foot (I didnt know about it) and was in pain - so when I was brushing him and brushed his leg, he snapped and bit my arm. That was a cairn terrier. There was a yorkie that had a sore in his mouth and had just eaten, so when I went to pet his head, he attacked my hand. Another dog (black lab) bit my leg when I accidentally let a door tap his behind when we were coming in - his glands in his bottom were swollen and the door tapping caused additional pain. I continue to work in rescue (though admittedly, not as actively since getting divorced - I'm gone (at work) long hours now (used to be a SAHM)and dont have the financial resources that I used to).
good luck.... I hope you are able to find a new home for your furry family member, and that your neice and child don't develop a fear of dogs or other animals as a result of their experience.