i probably shouldnt say this, as i am sure that the dogs trust do a great job, i just have never found their policies workable and dont have much time for them tbh. Yes, things don't always work out, it nearly didnt for us, and without continued support from Battersea we woould have given up.
I do hope you do get another dog one day, breed rescue is often a good way to go as they often come through foster homes so their temprement can be assessed. When we got our dog from battersea we were grilled for ages, they didnt do a house check as we lived quite a way away but they based their decision on my experience as an owner and vet nurse. Even then they were very specific about what sort of dog they would let us have, or basically, they wouldn't let us leave until we agreed to take yazz, who we had to go back up to see as he was at a different centre. He was a lovely dog, not for everyone, rotweillers are weird and have their own set of rules. I found them easier to train than my little terrier pup, he is a total loon, chews everything, totally mad, but its ok because he is small, but more importantly, knows just how far he can go with DD to avoid her getting upset etc, he is my little star.
Try not to feel bad, yes DD will be upset, you dont say how old she is, but she will realise that it was the best decision. You know you have done the right thing, i think it was very brave for you to admit defeat and give the dog the chance to find a more suitable home rather than hoping he would get better.