Given the extent to which a poster has misrepresented my advice, let me set things out very clearly.
OP and his ex both have parental responsibility. That means they are both entitled to a say in their daughter's education. They should attempt to agree which schools will be named on their application for a primary school place. However, if they cannot agree, whether by direct discussion or mediation, it is open to either side to seek a Specific Issue Order to determine which schools should be named. I am not recommending that the OP does so, nor have I ever recommended that he does so. But, if the parents cannot find any way of agreeing on the choice of schools, it is the mechanism available to either parent to resolve the conflict, with the courts deciding what is in their daughter's best interests.
Contrary to what another poster appears to think, parents can and do apply for and win SIOs without legal advice, even when the other side is represented. However, just to repeat, I am not recommending OP does so. He should try to agree things with his ex and only go to the courts if all else fails.
OP could, of course, agree to his ex's choice of school and her intention to apply using her parents' address. Some on this thread appear to think he should do so. However, he needs to be aware of the consequences.
Applications for school places must use the address at which the child lives. If an application uses any other address is will be treated as fraudulent if the LA finds out. It doesn't matter whether it is a deliberate attempt by the parents to fiddle the system or a misunderstanding. The parents may genuinely believe that it is ok to use a future address, but the LA will still treat it as fraudulent. It is, of course, highly unlikely that the parents will be prosecuted. Such prosecutions are extremely rare. I am only aware of one successful prosecution.
What happens if the LA discover that an incorrect address has been used before places are allocated?
If the mother's parents and the FMH were in the same LA, they would simply use the FMH as the daughter's home address. That would give her little chance of getting into schools 20 miles away. If she was not allocated a place at any of the preferences, she would be offered a place at the nearest school with places available. That could be some distance from the FMH and could be even more than 20 miles from the mother's parents.
In this instance, the mother's parents and the FMH are in different LAs. The LA could still use the FMH to determine admissions, or they could reject the application completely. Even if they processed the application using the FMH, if OP's daughter did not get any of the preferred schools the LA would not be under any duty to offer another place as they are not the daughter's home LA, leaving her without a school at all on offers day.
What happens if the LA don't find out that an incorrect address has been used until after places are allocated?
In this situation, the LA is likely to withdraw any offer that has been made based on the incorrect address. The place can be withdrawn even if OP's daughter has started school. This happens to a few hundred children every year, usually in the autumn term, but occasionally later in the year. The parents would then have to find a school with places available. This is likely to be an unpopular school that may not be convenient for either parent.
It is, of course, possible the LA would not find out that an incorrect address had been used. That would be the gamble OP would be taking if he agrees to his ex's plan.