M.;
I too, can completely understand your situation as I also have lived it. It is sort of nice to know you are not alone in this world as those that live with someone with Bi-Polar do not always understand the extremity of the situations one can live in.
I had a five year rift with my own mother and I am still not very close with her. I do not speak with anyone on her side for other reasons. This brought big bags of guilt when I lost my great-grand mother and then my grandmother as I was close to them. (I also do not speak with my natural father - he was given three chances - one more recently, which he completely blew!)
Now, I have a son that will be four years old. It is hard as he does recall visits with "Grandma and the horses". (My son has a memory that I would LOVE to have!). My husband could care less for my mom and her side of the family. This does make it difficult, but in all honesty he does at least tolerate my mother.
As someone that experienced much of something that is rather similar to you; I can only recommend forgiveness. If your mother is truly bi-polar, she cannot help her disease. If you know she is sick, then yes - this is a reason, but you also understand that something is making her this way.
Explain the situation in a loving way to your children. My step-father shielded my younger brothers from my mom's erratic behavior, but how can you hide the fact that your mom is holding your father at gun-point? Or the crazy mood swings? This is something that needs understanding before it damages your children's emotions to any degree so that they understand. It took me years to realize what was going on until I learned the diagnosis. Not that this makes it better, but it does make it easier to deal with. You at least understand it is not you!
Again - it is not you. It is not your husband or your other children. Your mom is the one with the issue and needs the love and the support right now. As hard as this is, that is what is going on and she may be getting this from your other daughter (Sydney) and she may just see something healing in her. Your daughter needs her, too. She may also be able to 'pave the way' if you are open enough with your children on this matter.
Best of luck!