As I understand it, the girls fell out at school but played fine when they were together at your place after school?
In which case, the mother won't have known about the school drama until she picked her daughter up from gym. So she wasn't "using you", you did a favour arranged before the fall out.
Not does it illustrate that the friend was fine. What choice did the friend have but to behave until she could confide in her mum? She isn't going to tell you all about how upset your daughter has made her, is she?
And kids don't work in a linear way. In fact, on the whole, people don't. Sometimes bad things happen and we're fine at first and then we fall apart.
My 12yo had someone pull a knife on him but came home all chipper. Wasn't until next day he texted to tell me. Explained it made him feel sick so had just tried to forget about it.
Also. Don't text unless to say how sorry you are to her this and you really want to understand what happened so you can talk to your child about it.
If you do the hugely long-winded and passive aggressive texts suggested above you'll make things worse.
Focus on resolution. I'm sure you're angry and that your child is devastated but try not to make it worse. Try to understand the other child's upset and work through the problem. Maybe the child and mum just need to be heard.