@Claireeo you need to do the right thing by your own child, not someone else’s child.
You have reasonable grounds to suspect that a fraudulent application may have been made. You don’t have to know for sure that it was, and not having absolute proof would not make your report malicious.
Tell the LA what you do know or suspect and leave it to them.
If the other child has legitimate reasons which give her preferred priority over your child nothing will happen.
If the family was genuinely living at the other address when the application was made, nothing will happen.
If the mother was gaming the system, the child will almost certainly lose the place but it was a place which should never have been offered to them in the first place and any distress to the child or the mother will be 100% down to the mother’s choice to make a fraudulent application and not for you to berate yourself about.
You can only do the right thing by you and your own child. If someone else has done the wrong thing, then any stress an investigation causes is on them. If they haven’t done the wrong thing, they have nothing to fear about the LA looking into the circumstances of their application and will be able to provide the corroborative evidence required.