Why would you send your little kid to a painting-themed party with a very expensive dress? And what's even worse, why do you think you are entitled to get paid for the damages when you were the one who sent the girl dressed like that?
People are weird, but parents are weirder. Being a new parent of a little kid must be really hard, and I don't doubt that at all. I may not have kids, but I'm sure it is hard to take care of them, especially when they are under 10 years old. It is a little kid who wants to know and discover everything but still knows nothing. That is definitely not an easy task to have to take care of. I will not be judging a parent for their parenting style. I'm no one to do so, and I don't think that is correct. Parents do what they can with what they have and what they know. Of course it's going to be hard, and it will involve a lot of mistakes along the way. This story is not really about parenting; it's about entitlement, and not a little kid being entitled, which is understandable and fixable. A full-grown adult entitlement that is cringy, pathetic, and concerning.
You cannot make other people responsible for the mistakes you make, and that is something that some people need to learn with urgency. If you decided things were going to be done one particular way, and it didn't work out. Then who do you think is responsible? The person who planned for things to be like that? Or someone else who had nothing to do with the planning but happened to be there at the time of the downfall? I think it's obvious that the person with poor planning skills is the one to blame. But an entitled person does not see things that way. If a 'Karen' went into a shop, bought something she thought was meat but was actually vegan meat, it had a label; she just didn't read it. Whose fault would that be? Karen's, of course, but Karen won't realize this, or would simply not care. She will go back to the shop, make a mess, and demand not only the money back but also some extra for her suffering. That's just another example of how entitled people behave.
So, going back to this mother, who do you think is at fault for sending the little girl with an expensive dress to a painting-themed party? The person who is in charge of her and dressed her? Or the venue owner who wasn't even at the birthday party? It's the mother's fault. That's why she needs to deal with it instead of trying to blame other people for her awful way of planning things and her complete lack of thinking skills when deciding to dress the girl for the party. If she wants to be mad at someone, she just needs to get mad at herself. After all, it's her fault.