OP, you mean well but your boundaries are very poor.
I think you should focus on the message you are sending your OWN DD.
I'm an adult and I'm finding your priorities confusing.
Your child is being treated badly in school by this girl that she has had issues with before, who impedes her making new pal's at the club.......yet you are continuing to bend over backwards to accommodate this other girl, against your own child's best interests.
I think it is so important to advocate for our children, to have their back, to let them know that they are who is most important to you, especially when they are such a great child, as your daughter is.
Why the hell would your child want to be in the car this evening with that little madam who has shunned her all day.
Honestly OP, please put your child ahead of your own ego in this matter, which IMO needs to feel "honourable", and is what you are prioritising.
I mean the above with kindness.