You don't mention a state or location, but as someone who has worked for several government agencies with "tipster hot lines" to report fraud and particularly children's protective services, whatever its called in your state, I can pretty much assure you that under no circumstances does a person have a right to "know who turned them in!" Usually that's the first question they ask.
When a person calls in and reports suspected child abuse, they do take down the information about you, but that information, AND the disposition of the report or complaint are things that do not get further passed. The person called in on doesn't get told who reported them, and the person making the call doesn't get called back and told what happened as a result of their call. The way people find out someone has reported them, 99% of the time is that the person who did brags to friends, "I called in on that witch! I've fixed her!" and it gets back to the person.
I suggest that you break off all contact with your ex except for the formal and documented communications ( I suggest by messenger services) necessary for the relationship with your daughter to continue. Drama junkies love to involve others in their drama, enjoy it less when they can't talk to you and argue with you and listen to you cringe and deny and get involved with their accusations. Failing to acknowledge this person's hogwash is the best thing you can do if it is not directly related to something related to your daughter's needs.
