OP, I would have a quiet word with the teacher.
If things are as you describe, suffocating for your daughter, then you need to give the teacher the heads up.
Absolutely kindness is very important but your daughter's refusal to go to the party is a clear message to you that she is being imposed upon in school.
IMO there is a big difference between being kind towards a classmate and allowing them to control and manipulate your child through her tears and demands.
My youngest had a friend that ended up in a dynamic like that with another child and it ended up very poorly with school refusal, a real mess.
Speak to the teacher.
Your child needs your support.
Ask the teacher to intervene and not allow your child to be deprived of choice because this child is fixated on her.
I appreciate this is difficult, but I would be listening very carefully to what exactly is going on in the class.
Sometimes busy teachers can accommodate a demanding child despite it being not in the other childs best interests.