I think this is the point to back right off, let the teacher handle it, and politely say to the mother that as you have such different parenting styles you think it's unfair on the kids to get caught up in it, so the lift arrangement isn't going to work. Reassure your dd but tell her this will blow over if it isn't turned into a big drama, to be pleasant and polite but not to try to be friends with the kid right now, wait until the dust settles and focus on other friends for now.
The kid is 6. It's human to be enraged with a child upsetting yours, but the real issue here is with the mother, and her lack of boundaries, and not the poor child being wound up like this. It has the potential to blow up into something it doesn't need to be, which would make things hard for both kids for a long time to come. I'd just focus on soothing your dd and reducing the situation to size - her response will largely mirror yours.
DS has a friend he's known half his life, who has another friend, in their class at school, who is an evil little brat (clearly an appropriate way to describe a 5 year old, yes) who does stuff such as run the length of the playground at hometime, when DS has fallen over and hurt himself and is crying in my arms, to tell him he can cry harder now because DS' friend is coming to his house, and DS is not invited. Said evil little brat is only just 5. He is also spoiled as hell and runs to mummy every single time anything goes wrong, and she keeps him home "when he is sad" if another kid is "mean to him". They are, sometimes - one told mine he'd shoved the other out of a door, assuming mine would think it was funny, but he hates conflict and told me "that was naughty too, wasn't it?" At the moment, mine is being told he can't play with the vile brat, King Wasp (who orders the rest around) and his friend unless he eats snacks they approve of. Bananas are not allowed and nor is dried fruit. DS was asking me to ensure his snacks were on the approved list, and it took all I had after my initial 
and 
to cheerfully tell him no, I wouldn't alter the menu, and he was just to say, "Oh dear, that's a shame, I'll have to play with N and R today instead, as I like bananas" and to walk away. DS, seeing my original horror, had asked if he should talk to the teacher... but the thing is, that would escalate it, and also not teach DS how to manage things in an assertive, non-victimy way. He needs to make it clear that he's nobody's doormat, but he also needs to be calm and not turn it into a war, as these kids are currently frenemies, at worst. So I explained that he could just make it politely clear that he won't be told what to do by other kids, and will just walk away from efforts to make him, but not to turn a silly thing into a big deal, and to play with them happily if they weren't telling him what to do. Which is what's happened for the rest of this week, it seems. But though he's happy enough now with them, I'm still spitting chips on the inside.
I suppose the point of that rather garbled story is that our own protective anger is inevitable, but I think there's a real risk in reacting in a way that soothes those feelings, and doesn't help the kids work it out. I swallow down dislike and anger with evil little brat when socialising with his mother because any real dislike and hostility will inevitably affect their situation, and for all I know they'll be best friends in a year or two. Kids this age change so fast and the politics is so endless, and we have to try, somehow, to give them the tools to negotiate those waters, because we can't be there with them.
Sorry if that makes no sense. It's something I am having to work so hard on, myself, because I am angry to see my sweet, kind kid be told what to eat and who he can play with, and while I am trying to equip him with the tools to handle that diplomatically, while being nobody's punchbag, my instinct is to ask the teacher to stop it (I may still have to, but I so hope not) or to tackle the parents. But that is about my own hurt at the behaviour, and wouldn't help them.
I recognise that this mother has created and is fuelling the situation with her own protective mother instincts being so out of whack with proportionality, so it's rather different. But I do think it's easy to let our love for our kids get in the way of detachment, and thinking about their long term best interests. My mother always used to charge in to fight my battles, and it did me no favours - in the end, I stopped telling her my problems. Which was a very lonely place to be.
I hope that isn't patronising and I'm I'm sorry if it sounds it. (I'm also very sleep deprived and hormonal, so lucidity and tact is low!) It's just something I am rather painfully working on myself right now. It hurts to see your kids deal with this, doesn't it. DH said to me the other night, "I don't like having kids. It hurts when they're coping with this." I'm sorry you are, too. 