Cutting off a selfish sister

I need some advice. I had my second baby just over a week ago. I have an older sister who has 2 teenage children. A few weeks ago at my mum's house, her son and husband were going on about how I couldn't get my first child to sleep. I snapped and said I'd leave then so she could sleep in the car! To which my sister snapped fine go.

Before this, we had stopped speaking for a month because she didn't come down for my birthday. I think there were other reasons behind this, but I was really upset by this.

So now to when I have had my baby. I message her but all I get in result is "Congratulations x" Not even asking if we're ok. I held my tongue and didn't reply with what I wanted to. Fast forward to a few days ago, and we had a blazing row over text. She has held onto the comment I made about leaving and termed it as me threatening her son! Amongst many other things, I can't see that I want her in my life anymore.

I've been there and done so much for her but I'm now done with her selfishness. I need advice on how to emotionally cut myself off from her so I don't continue to get upset by her, but how to do it so my mum doesn't get caught in the middle. I can't deal with her and how she has made what should be such a happy time in my life about her and her grudge. She said some really really spiteful and hurtful things that I can't ever forgive her for, especially at this time in my life and I'm sick of always giving in. I feel bad my mum is stuck in the middle but I don't consider her my sister anymore. I just advice on how to get through this.