OP, if you are still reading, well done to admitting that you might have got this wrong. If folks seem harsh, that is because you posted in AIBU.
Try posting again in Relationships, saying that your MIL is coming to stay for a night, which you are dreading as you don't get on. You find her judgy and her comments undermining. However, as you lived with her for 6 months,, and she is your partner's mother then you know that you have to get on.
You might get some good advice there - like, yes, do invite her, but ask your partner to explain that you will all have a nice time if she is able to leave you guys to do the parenting this time. You value her, you love her, you love the duvet she gave you, but you find it difficult if she corrects your parenting especially in front of the children. Something like that?
Best of luck with the birthday. It is stressful when you have an image in your head of how you would like a special day to be, then someone or something comes along and changes that plan. However, you can still enjoy it - and remember, it is the first of lots of happy birthdays. I spent my ds1's first birthday throwing up. My husband was away at a conference and my mum and dad who had come to stay was immunosuppressed at the time, having had chemo, so she had to leave us at 6.30 am. We have some nice pictures of a baby with a birthday cake, taken at 8pm, when dh got home (woke ds up to take them!!)