Sounds like the daughter is doing her level best to discredit you, make dad angry at you, and he and mom will get back together. It's a typical ploy kids will do because their security and sense of continuity and familiarity has been disrupted. The mom can only take in what the daughter tells her. The kid's world is upside down. If mom starts not believing her then she has nothing to grasp. She needs mom's confidence and approval because it's some way she can make sense of things. It's something to hold on to. Wrong as the accusations are to put forward, the girl is trying to feel some control over something that unwound. She unfortunately sees you as the cause, because you're the only factor that's been different. It's the working of the child's mind. It doesn't always make sense.
But definitely talk to a lawyer. Defamation of character and false accusations usually don't go over well, legally. Slander doesn't either.