Wow this sounds familiar, as my DD is 11, but she is personality a lot like your DD. And I do think the sensitivity gets picked up on, and girls with mean tendencies can smell blood, and go under the radar to incite drama. I live in a small house with thin walls, I have heard all kinds of snarky-mean behavior. You can lay it out on the table for the Mom if you want to. Been there, done that. I'm convinced the only person who will decide if she will treat your daughter kindly and like a true friend is that child herself. If that child doesn't value her friendship over her power and control, they won't really ever be close friends. Moms cannot force the issue, it doesn't work. Your daughter has told this girl what hurts her feelings, but it persists. Mean girl games. I'm so sorry this is happening.
Here is what I would do. Tell the Mom you need need a little break this summer from the regular playdates you've been having once a week for 2 years. It's the end of the school year anyway, and a natural time to transition out of routines. I would keep it light and not go into your real reasoning. You'll be busier, etc. Talk to your child's teacher about what is going on, and request they NOT be placed in the same classroom next year (although you said they aren't this year, I would be proactive to keep it this way). Keep your daughter busy with fun activities and other playmates. Stop scheduling playdates with those you are friends with personally and always ask your daughter, "who would you like to invite over to play?" and follow her lead. Schedule your own time with your own friends separately, mom's nights out, etc. Talk to your DD about what she likes about her peers. How she feels after spending time with someone. Empower her to make choices in friendships. She should not ever feel stuck in this mean girl's company. If they invite her over to play, don't accept right away. Make time to ask your daughter if she wants to get together with her. If she doesn't. Say "I'm sorry, but X doesn't feel up to playing this week." The little girl learns that she does not get to pull all of your DD's strings. She may rule the roost at home, but she'll see that your daughter will be off making other friends if she wants to act like a mean bully.
Your relationship with the Mom and the family will surely change, you may drift apart, but I think your daughter will be a stronger, healthier person for it.