I think you need to take a massive deep breath. This is all perfectly normal friendship stuff and you’re possibly projecting your own issues onto your dd here. And it doesn’t get easier. Wait til you get to Y5 and 6 and beyond. Do you work and have friends of your own to cultivate relationships for yourself and her with? Mums who are happy and confident in their friendships model that for their children.
The birthday invitation thing is a bit rude and weird. What reason did she give? Did she invite the wrong “Sarah”? I do that every year and get Joey and Harry and Louisa from the wrong class. I personally have never uninvited anyone, but without understanding why, it’s hard to say.
All that being said, is your dd actually a good friend? My youngest had a “best friend” in reception and Y1. He was an absolute nightmare. Rude, nasty, swearing, mean. My ds was quite shy and liked everyone so this boy just sort of claimed him as his friend. Lots of play dates at his, I tried to humour it, but I never invited to ours. I didn’t want my house burnt down. His mum thought they were absolutely best friends and he is an angel in her eyes.
He eventually moved away like 200 miles to live with his dad, but every school holiday he’s back and I get loads of messages off the mum. They do creepy things like turn up at the playground near our house (the mum lives like 20 minutes away so they come here specifically to look for us). Last year they came trick or treating here and were wandering around the village looking for ds. It’s like let the friendship die, your child is a little ruffian, you’ve moved 200 miles away like 2 years ago now, stop stalking me. 😩
That’s not to imply it’s the same situation, but maybe the friendship was one sided and not good for this other child. She’s letting you down as gently as she can. I think you need to take the hint and help your dd to cultivate new friendships and social networks.