i think if you make her do a MA she doesnt want to do it will turn her away from all MA and make her eventually resent you. i think your best bet is to use this style as a stepping stone to something more useful. shes a little girl so she probably likes this bc it looks like dancing. but at least its dancing with some concept of MA. if you look at it in the long term its better for you to let her and then try to redirect her into a more useful MA later on. your wife is happy and thinks you are understanding, your kid wont hate MA just bc her dad made her do them, and eventually if you can get her to switch to Kungfu or karate you get what you want too.
i would not push her towards judo though. its a fun sport but on a more superficial notev you start a little girl off on that by the time she is in high school she will have shoulders like a linebacker. just a thought.