From everything that you've posted, your sister sounds really unwell to me. She sounds very toxic. I would not want to be around her if she was my friend. I am not sure what I would do if I was related to her. She is crossing so many lines it isn't funny. She should not be sending you emails saying that you are a bad mother.

Has there always been this pattern where she says horrible things about you and your family puts up with it? Does the golden child/scapegoat pattern seem familiar to you at all? Does she have these kinds of problems with other people? I am assuming so, since she told your family to rot in hell.

You are damned if you do, damned if you don't. What you are doing now seems like the least damaging route. That you even tolerate her presence at family gatherings is very generous of you. If you bend to her will and apologize when you haven't done anything wrong, she will just learn that she has to be cruel to get what she wants out of you.