advice regarding 10 y.o

So, my s.daughter just turned 10 about a month ago. Her mom told her at some point that when she turned 10 she could choose what she wants to do (where she wants to live). Also told my husband and I a couple years ago that we'd talk about doing 50/50 parenting so we'd have said daughter every other week and she would sign off on child support as long as we'd still help out financially with medical stuff and some things outside of medical. We have absolutely no problem whatsoever helping out. We've paid sports fees and bought whatever equipment was needed on more than one occasion, mom said she'd split it with us (it got expensive) and we never saw a penny back for it. Paid school fees and have bought some school supplies. (Husband is only supposed to help out with medical [according to court papers] and anything outside of medical is considered a gift unless otherwise discussed). Well, s.daughter was talking to me the other day and said, "my mom STILL hasn't said anything to me about week to week..and she said when I turned 10 where I want to live. I don't even want to do week to week, I just want to live with you and dad. My birthday is here and gone and I haven't heard a word from her! Do you think maybe she's going to do it for a birthday present and just surprise me?" I told her I have no idea and maybe she could just mention something to her mom about it. She said she would day we got her, she said she talked to her mom. I said oh yeah? How'd that go? She kind of sighed and said, "well, she said she doesn't know yet. Even though she told me when I turn 10!" I told her I didn't know what to tell her other than maybe she could gently remind her what she said. Well, s.daughter then asked, "can you and dad just go to court and I can tell the judge I want to just live with you guys and not my mom? I just really don't want to live with her anymore! It's just not fair that she can just change her mind like that and lie to me."
Does anyone know how this works and what our options could be? The first time she ever just came out and told us she wants to live with us is when she was 4.