Firstly well done for ending the abusive relationship, you did the right thing for yourself and your baby.
For the sake of clarity I am going to use AX for your abusive ex-partner, and LF for the baby's legal father (the one who you think is the biological father and who is on the birth certificate).
It's not completely clear from your post but I think you are referring to AX for most of it, and the two of you haven't been in touch at all - presumably because of the restraining order and the upcoming court case.
I can't work out whether you are still in contact with LF, presumably he must have registered the birth with you (since he's on the birth certificate) but is he involved at all - whether in a relationship with you or just involved in the baby's life? Is he paying child maintenance?
Personally I would want to ask LF to do a DNA test to be absolutely sure, for peace of mind.
AX might not be interested in the child, but if he is and he asks about it (directly or via lawyers) you will have to tell him at that point that you're sorry but you got the dates wrong and he is not the father after all.
FWIW I think you are right to follow the advice of Women's Aid as they're the experts and presumably they know more about your situation/history than what you've posted here. So if they advised you not to tell AX that he's not the father, please trust their advice over the opinions of strangers (including mine!)
Do you have access to legal advice/representation? Depending on your finances you might be eligible for legal aid (due to the domestic abuse). If so your lawyer should be able to advise.
Best of luck with the court case 