I agree wholeheartedly with LucyJoy and other poster's here.
Having been the victim of many triangulations, I've found that it often pays to 'circle the wagons' - e.g., to touch base with the other Mom and let her know that you appreciate her giving your daughter rides to this activity because you really do support it, but to also let her know to end the rides if your daughter exhibits any incorrect behavior.
I know, this is especially tough to do with a parent you don't know - some parents I've talked with have reacted well to this and others not so well, but I've definitely found that not doing can lead to other problems down the road.
Take the break, but I would also consider not having it be too long, as then your daughter gets what she *thinks* she wants - you out of the way & a new person to convince that you are mean (or whatever) to her.
Reaching out to this Mom sets limits to what your daughter can do to manipulate, which is always a good thing, as she will know that you will not hesitate to let other parents know.
I call the parents that own the home when my HS kids plan to hang out outside the house. At first I got lots of protests & fighting on it, then just complaints, and now my kids call the parents for me & hand me the phone.
Best of luck from someone who's been thru the ringer right there with you!