I agree with those who ask 'where is your anger'?
Enough already with the poor neglected, child story, you have no idea if this is true or not. Your posts focus on this aspect and I am guessing you are proud of seeing yourself as #bekind type of parent.
This bully girl is not abusing your child because he has a difficult home life. She is bullying your dd because she can. That's it. Because she can. Otherwise she'd be bullying the teachers and children older than her but she isn't. She is bullying your dd because she thinks your dd is meek and an easy target. And she is bullying your dd because she has no empathy. Not all kids, not even most, who are abused or witness DV (you don't even know if this is actually the case) bully their peers. This girl does because she finds it convenient to use your dd to feel powerful, she has a nasty personality. People who pick on those they deem weaker or meeker than them are despicable creatures and at 10/11 this girl know what she is doing.
By rationalising why this girl is being horrible you are undermining your dd, I feel sorry for her. While you are polishing your halo, as you are oh so able to see all sides, your dd will end up feeling guilty about hating or at least disliking the girl as her mother tells her this girl has a difficult home life. It doesn't matter, there no need to look for a reason in why this girl is doing anything, look at how your dd responds and what she can do to defend herself. IME, these bullies never pick on strong, athletic loud mouth type of kids, no it's the quiet, kind and physically smaller or slighter / slimmer ones.
I would tell school they are failing in their duty of care and are not safeguarding your dd. They need a 1-2-1 TA to keep a close eye on her at all times, although by the sound of it, this girl would still find a way to bully, it's how she feels powerful.
Work on your dd's body language. Tell your dd to stand tall, hold her head high and lift her chest at all times in school. Body language is a more powerful communicator than words which the bully can turn around and use against you. Let your dd do a sport, not necessarily boxing but that's ok too. Running, team sports. If you have no way of avoiding the same secondary ensure your dd joins team sports if she is sporty. If needed, she can talk loudly at bullygirl and make fun of her a few times.
Bullies do pick easy targets. Don't let your dd be an easy target. Telling her about this bully's supposed difficult background is not helpful.