Aibu not to encourage this friendship?

[sunbeamsandsparkles]

DD has a friend in her class, they are both 6, who she has been friendly with since starting school.
they have quite a turbulent friendship and DD often comes home telling me that this friend has hurt her or said something mean. they are usually back to being friends the next day etc but there has been a few times that DD has been very upset about things this girl has said or done and I've had to speak to the teacher. DD has lots of friends in her class who she gets on really well with and who don't make her upset and I've been trying to steer her towards them instead over the past few months.
anyway it's now the summer holidays and this girls mum has messaged me a couple of times already to try and arrange play dates, we've been busy so I've managed to just put her off for now but she seems really keen and says her DD keeps asking to see my DD.
aibu to say sorry I don't think a play date is a good idea? Or do I just let DD figure this out for herself?
DD would happily have this girl round for a play date.
Her mums quite a popular parent with the rest of the class and I don't want to put DD in a situation that's going to mean this girls behaviour may get worse towards her but I also want to show DD that she can surround herself with better friends who care about how she feels.

I’d absolutely have her round and watch what the dynamic was, what’s actually going on when they play, why it seems so fractious.

YANBU to be cautious, but I'd think carefully before flatly refusing a playdate outright — a few things worth weighing:
Why I wouldn't hard-refuse:

• DD herself would happily have this girl round, which matters. At 6, friendships are genuinely turbulent and inconsistent almost by nature — that's not necessarily a sign of a "bad" friendship, it's often just what 6-year-old social dynamics look like as they're still learning to navigate conflict, share, and regulate their own feelings.
• A one-off supervised playdate at your house is actually a good, low-stakes way to observe the dynamic directly yourself, rather than only hearing about it second-hand from DD after school. You'd be right there to see how they interact and step in if needed.
• If you refuse outright, DD may feel like her own friendship choices are being overridden, which could make her more drawn to the girl (forbidden fruit) rather than less, especially if she doesn't currently see it as a problem the way you do.

Where your caution is fair:

• You know your DD and you've already had to speak to the teacher — that's not nothing, and it's reasonable to protect her from repeated upset.
• You're allowed to shape the environment without banning the friendship outright — for example, keeping playdates short, at your house rather than theirs (so you can observe and step in), and not overcommitting to regular ones.

A middle path: you could say yes to one shortish, home-based playdate, framed to yourself as information-gathering, while continuing to nurture DD's other friendships in parallel. That way you're not the one who "banned" the friend, DD gets to make her own social discoveries with you nearby as a safety net, and you keep the option open to gently deprioritise it later if things go the way they have before — without DD feeling controlled. I'd lean towards: let her have this one, on your terms (your house, short duration), rather than refusing altogether. YANBU for your concerns, but I don't think you need to shut it down completely either.