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The short answer is that you are unlikely to face any issues due to this.
Now for the why? The police will not continue to investigate this or be involved with it. If anything was going to happen legally (in a criminal sense), it would have happened that night. I think that the police probably realized that the other parent was being contentious and let it all slide. They were not required to give you Miranda rights at that point because it was purely investigatory/fact-finding. The reality is that the probability of criminal charges is, as you said, thin.
On the civil side, the father could always file suit. However, I don't know very many attorneys that would take this case on an contingency seeing as there are literally no actual damages (as the child was not physically hurt). Most attorney's would probably require the father to pay for everything out-of-pocket which would be highly expensive (most attorneys charge about $300 per hour on average). Most people will not take it this far. I seriously doubt this would make it to court or even survive in court if it did.
I hope this helps!