This mom has issues, and she is projecting it all on you, while at the same time using her daughter as a pawn. None of it is right.
However, I was with you all along until you had the sleepover and you did not invite the little girl. I think that it would have been a nice gesture to demonstrate to your child that while she and this girl cannot be friends, they can be "friendly," and this can be seen as a gesture of good will. I understand that at times friends grow apart, but it is clear that this little girl did not want to stop being friends with your daughter. I am not sure is a "clean break," meaning, going from being invited to suddenly being "uninvited" to functions, was the right way to go. It is hurtful. It may have helped to include her still, to some functions, until it becomes evident to both girls that this friendship has run its course.
Now that this mom has lashed out on you in such a public way, there is no turning back or fixing things...she's got a vendetta going and plans to seek her revenge or you and possibly your daughter as well. Awkward.
What you can do is talk to this mother in private; honestly, when dealing with "off the handle" folks like this it is a good idea to just apologize. Tell her again that it was not your intention to exclude her daughter, and in the future, if she is willing, her daughter is invited if she desires to come. At the same time, let your daughter know that this does not mean she and the girl have to be BFF's -- it is only a gesture of good will and being kind.