Oh OP, your hurt and bewilderment at your mother's behaviour comes across loud and clear, and it's shameful that she makes you feel like this. I totally understand the grief this triggers - knowing that the relationship you feel you should have with your mum simply isn't your reality.
My situation is a little similar. I'm my dad's only child, and yet he still manages to have a golden child that isn't me. While he may say the right things at times, his actions show the exact opposite, even to the point where he is getting married in a month's time and has chosen to go abroad with his partner and her adult son and his girlfriend, but there was no invitation for us to join them, though they did invite other friends.
There's no getting away from it - it hurts, and I think it always will. I have spent years grieving for the relationship I wish I had, and it hurts that he's not the grandfather I'd hoped he might be. I can't imagine the pain your mother causes by behaving so very differently to your brother and his family. It is totally unacceptable.
It's all too easy to tell you to stop caring, pull back, and do less, but the reality is it's hard. But, in my case, it has honestly helped. I have chosen to focus time and energy on the relationships (friends and family) that genuinely lift me and make me feel good, and he is now more distant. As a result, he has a little less power and causes a little less hurt. I still see him, but on my terms and when and how it suits me, and this has helped enormously.
It sounds like you and your sister have each other for support, which is amazing. This type of upset can make you feel very alone, but the fact you have each other and understand completely what you're both feeling is enormous so lean into that. Try not to exposure your children to this and start pulling back, little by little, so you see her less and her grip on you will get smaller, I promise.
Most importantly, remember, this isn't up to you to fix. By inviting her to things, involving her, hosting Christmas and being there for her whenever she needs you, you're indirectly telling her that her behaviour is just fine. She doesn't deserve this, and while it will be hard, maybe it's time to stop being so warm, kind and inviting. Let her reach out if and when she wants or needs to. Let her take on some of the responsibility for your relationship, she's got away with not bothering for long enough.
Sending you love, I know how painful it is, but redirect all that love into your own four walls - they love and adore you so you'll get it back in spades.