I have never described her as an "outrageous fraudster". The use of the term "fraudulent application" for an application using an incorrect address comes from the Admissions Code. No misogyny involved. That is yet another straw man from you (and yet another ad hominem attack). I have said the same about ex-husbands who have proposed making fraudulent applications for places.
The question for the application is not where OP's ex lives. That is irrelevant. It is where OP's daughter lives at the time of application.
Currently, OP's daughter lives permanently in the FMH. If that were to change but she was still spending her time 50/50 with both parents, most LAs would insist on the application coming from the address of the parent who receives Child Benefit. OP tells us that is currently the mother, but it is being transferred to him. He also tells us that the current living arrangement (50/50 with each parent but with the daughter permanently residing in the FMH) is mandated by CAO, so his ex should go back to court to get the order varied if she wants to change this.
OP's specific question is whether his ex-wife can use her parent's house as the address for applying for school places whilst their child's registered address is the FMH. The answer, as I have said, is no. That would be a fraudulent application.
Even if OP's ex has moved with her child to live with her parents by mid-January, that is where the child is living for the majority of the time and she is not in breach of any court orders in making such a move, she would still face a further problem in that most LAs now insist that, if a parent owns or part owns a house, that address must be used when applying for a school place even if they don't live there. So, unless her parents' LA is not one of them, an application using their address would still be fraudulent unless the FMH has been sold or OP has bought her out by then, which is highly unlikely.
If, by January 15th, the child is still living in the FMH or spending the majority of her time there (as OP believes will be the case), or, indeed, should be living there under the provisions of the CAO, any application must come from that address. If OP's ex still owns a share in the FMH on January 15th, it is highly likely that any application must come from that address.
It is possible that OP's ex might be able to arrange things so that an application in mid-January from her parents' address would not be fraudulent. However, even if she manages it, given the way these things work, there is a good chance the LA would still regard her application as fraudulent, leaving her and OP having to appeal for school places.
Trying to sort out school applications when parents are separated is often difficult. Even more so when they are in the middle of divorce. The system isn't always helpful in this situation. I have, sadly, advised parents who thought they were doing the right thing for their child, but who had unwittingly made fraudulent applications. At secondary school level it is sometimes possible to help them sort this out and at least get them into a better school (or one closer to home) than the school allocated. At primary school, however, in most cases there is nothing that can be done.