OP please get help. Your poor ds ☹️ he's 10 and this is his home life, being physically abused by his violent sister while his mum seems to cover for her. I'm not blaming you but this can't go on.
Yes DD needs support, that much is obvious. She is also almost an adult, who has been warned multiple times. She should be adult enough to control her anger. Yes she probably has issues but that is not an excuse to bully and harass her brother. Yes her brother hits back with words. He might not want to seem weak and this is the only way to get back at her. Your DD needs to learn this isn't acceptable. So far she's learnt that her outbursts are kept secret and lied about. If she is violent as an adult and to people who will not keep quiet, such as people in clubs or colleagues, she will be in a heap of legal trouble. Actions have consequences
Perhaps I'm biased as u worked in a children's home with an extremely violent yoynd person, but let me tell you, she was scary when she turned. And I'm 23 and a griwn woman, not a 10 year old boy.
Sending DD to relatives isn't punishment. It's necessary to protect and safeguard your son. Because if things Carry on and something serious happens, questions will be asked about what you did to prevent it.
You can arrange support, counselling, etc for your DD while she's living away. Breathing space might do everyone good. But more importantly, it will allow your ds to sit on the sofa or come home from school without worrying that he's going to get punched or choked.
Do it for Al your sakes op but especially your son, who is at risk of serious harm