I'm a bit confused as you say you don't have the time to ask for a review, but you also say the letter went to your housing aid legal representative who is asking for a review.
Presumably the council have asked for more evidence about the domestic violence - that's hardly unexpected. I'm afraid they have so many people seeking housing help they need to verify that what you say is true - otherwise anyone could roll up with a sob story. There should be no problem in verifying what you have told them, surely?
I doubt they would have sent a letter to the wrong address deliberately. If they didn't want you to get a letter they wouldn't have sent a letter out at all. Your housing rep got the letter - that's the important thing, as she is acting on your behalf.
I'm afraid you just have to go through the system. I know it must must be frustrating for you, but frankly I wouldn't bank on getting much help. You're in a safe, clean house. You aren't actually homeless, you just don't have your own home. There is a difference. Emergency housing is for people who have literally nowhere to live.
Re their suggestion that you apply in Northampton - well that's just their policy, in the face of huge numbers of people applying for housing who they can't possibly accommodate. It's not personal.
I can understand why you think Northampton isn't suitable because you want to live in Bexley to be near your family for support. But you are actually living with your family now, in conditions which are far from overcrowded, yet you want to move out. That doesn't really support your argument - do you see what I mean? You either need to be near your family or you don't. If you do, then you're already in the best place, surely?