Maybe she should be inspired to do her work in both cases. Then no matter where she is she will do it, and it will be useful when she lives alone, or goes off to college. I guess so far that isn't working, but maybe everyone could be patient until the kid learns self-responsibility that sometimes only comes with age or not at all.
Maybe a tutor would be useful. I think there is an added benefit when a child see's that school-work is important enough for outside help.
I think the kid has two different sets of rules from house to house and she needs to stabilize.
Like I said, I don't know what I'm talking about, I was a horrible
almost step-mom at one point. I was the only person that had boundaries and guidelines which made me look like the monster. Any rule I had was completely unenforcable and it's stupid to have absolutely no power in your own household. So I had to leave, there was no fixing it, and was going to be damned if I was going to be an impotent observer of insane doings in my own house. I hate the wars with the bio. mom so I will never be a step-parent. I went through the worst nightmare with that and never dated a man with kids again, because of it. (enough about me blah blah blah) I had to go to therapy because of that, anyway!!
I hope that you can keep the eye on the prize. I bet you are being manipulated by the bio. mom in more ways than one, and that might skew the point of things. If the kid saves doing stuff until she gets to your house so be it. You can train the kid, not the mom.
I know how frustrating the situation can be, as I said, I went through
the worst. I hope these situations turn out well for you, because there will be many obstacles, this situation is just a symptom of all the other fun to come. Just try to be fair and not punish the kid for the actions of the mother. I wish I could tell you not to let this crap get to you, but well, it should. It's just how you deal with it that matters.
Ok stfu Cicero...!!! lol!