L.,
I will say that I do not agree that braces are a privilege. If the dentist is saying she needs braces, then she should get them. At 12-years-old, it will be a lot easier to deal with braces... it's more common... than when she's a teenager or an adult. Braces are not cosmetic... they do straighten your teeth, and of course, that looks better, but they fix things inside the mouth as well that could cause bigger problems later. I never had to have braces, but I know from my friends who all had them, that they are not a fun thing. They are uncomfortable and a hassle, so I don't think they are something to be taken away because she's not doing her school work.
I do agree, that the bill needs to be paid so the 9-year-old can go to the dentist. That is a necessity. What does your husband say about this? Maybe you guys can talk to the mom about getting her in soon, to stay on top of her dental care.
As far as the school work... it sounds like there might be a bigger problem. Is she overwhelmed at home? Is she not managing her time well? She is only 12-years-old... that's still pretty young. Maybe you could talk to her about strategies that will help her get things done. If all she's hearing is negativity, she's going to feel like she can't come to her parents for help. She could be totally stressing out about it, but afraid to talk to you guys, so it just gets worse. If she feels safe in coming to you, and you guys can work out a solution (not just a punishment) together, then it might solve the issues. It sounds like she's got a lot going on in her young life, with a divorced home, trading custody, band, sports and school... that's a lot. Needless to say, she's starting to enter her teenage years, which from emotions and hormones alone can be so incredibly stressful!
I say just talk to her and see if you can get to the bottom of what's going on. It's probably a cycle that's just spun out of control and she doesn't know how to fix it. If you talk to her in a mindset of helping her figure out what to do, and figuring out where she's coming from, instead of telling her what to do and getting upset at her because she's not doing it "well enough," I bet it will go a long way.
I hope this helps!
Good luck with everything!
S.