At 16, I would expect my child my child to be able to own a decision they have made be it a mistake or not.
But I would also expect an ADULT to be able to check that the money exists before buying such a gift: send me the money first and then I ll buy it or something like that.
I would also expect that, unless they all have a part time job, the mother would contact the parents to make sure we were all in agreement.
Ultimately, the 16 yrs ild were just that teenagers. And you made a mistake. I think you are more cross at yourself for having trusted them in the first place including your daughter as you say.
The next step is up to you: you can suck it up or you can ask for the bracelet back and return it / sell it, ultimately it is a lesson you all have learned.
The other mother didnt have to be rude about it but she has a point. Sorry it happens though i know it is upsetting (she who just bought mega overpriced socks for her husband because they looked fun
)