a person who already has a bad relationship with the ex does not want to be pointing towards the ex for child abuse unless they are reasonably sure. The cost of making unfounded allegations is too high. Secondly, any person who invites CPS to invade their lives unless there is no other choice is an idiot. CPS can and often does ruin lives, to include the lives of people who did nothing wrong.
If the money is available without asking the father then perhaps getting the kid in front of a shrink is a good idea, but this can run into serious money, they are often wrong, and shrinks too sometimes destroy the lives of the innocent.
A kid who is five is darn well old enough to know right and wrong, is old enough to communicate bad behaviour of adults, is old enough to be the primary trigger for alleging child abuse. When I read the OP I notice that the mother and the father don't communicate, the mother alleges that the father and the daughter don't communicate well, and the mother does not communicate well with the daughter on at the very least several major important areas. Working on relationships and communication is at least productive, making unfounded abuse allegations has a high likelihood of being extremely destructive.