Without reading the other responses, this is my opinion....hope it helps in some way:
If this child is in kindergarten and acting this way, it is nothing new to the parent. SHE KNOWS her child and her child's behavior. I would definitely make sure that the TEACHER is sending a note home for EVERY incident. Sometimes teachers feel that when they address something at school, it doesn't always need to be addressed at home too. I don't necessarily disagree with that, but a poor behavioral pattern needs to be duly noted and the parent repeatedly advised. This way it is coming from the SCHOOL. Remember, your daughters could end up attending the same school throughout their whole childhood, the less animosity that has to be endured between the two of them, the better. If you feel like there is too much happening in the classroom, then request a conference with the teacher and one of the principals or the school counselor.
As for the party, I would respectfully decline. You don't have to offer any explanation. When she sees how many children DON'T show up, she may begin to understand. Or, maybe not. But until specifically asked if there is a particular reason you decline invitations, I would leave it alone. My daughter had a girl in her class in 1st grade that invited the entire class....22 kids....to her birthday and not ONE even showed. My daughter refused to go, and I didn't make her. You don't have to give your money and time to someone who doesn't treat you even so much as fairly. My daughter was sad when she came home the following Monday to learn that nobody went. She still felt sorry for her. Today, 7 years later, they are somewhat friends. They don't hang out together outside of school, but they do text every once in a while and such.
And, while you feel like a parent would want to know their child is a bully, they don't always want to know. And, sometimes, they don't see a problem with it -- we have experienced that too.
Pick your battles wisely as there are many to come. But, teach your daughter that the other child's behavior is wrong and that she is NEVER to behave like that. And, that she doesn't have to be friends with someone who treats her poorly. THAT is one of the best lessons in life.
My daughter had a bully one time that every time the girl would do something to her, my daughter would look at her and say, "Good Always Wins, Haily! Good Always Wins!" One time the girl got so mad at her for saying that to her that she slapped her across the face. So, I of course had to take on that battle, but after that day, the girl never returned to the summer program again. But, I couldn't help but find a small piece of humor in it (after a few days of cours, after my anger had subsided). If you think about it, there was no verbal response that would be able to contradict what my daughter was saying....what else was a bully to do?
As I have rambled, I have realized something very important for you..... this is only kindergarten -- it's not the last bully you will encounter. Make sure the one you decide to take on is the one that needs to be taken on..... you don't want to be known as the over-protective, over-reactive mother. And I am NOT saying that you are either one of those, BUT you don't want someone to wrongfully develop this opionion of you and then have them not take you seriously on a much more serious matter.
Anyway, I hope I have helped.
Good Luck!