Would I sue the Dept. of Social Services (the agency) for the sole purpose of obtaining the name of the accuser? Yes.
Don't most government agencies have some sort of protection whereby they cannot be sued? Absolutely.
Is it not their job to follow-up on ALL reports filed? Yes.
They have no way of knowing claims are false until investigating, right? Yes.
And that's exactly the problem with suing them - it is not their fault that someone filed a false claim. They don't know it's false until they investigate. Your suit would be solely for the purposes of gaining the information you need - to sue that party - but likely it will be dismissed before it ever got to the discovery phase, for the reasons above. I understand that it's frustrating - and you need to decide if it's worth the continued frustration and expense.
Is there not some duty they (the agency) have to "investigate" or at least contact the alleged abuser of the child? They do have an obligation to investigate and, if they feel it is appropriate, they reach out to the alleged abuser. If they didn't reach out to you likely they didn't feel it was credible. Or perhaps there is another reason. If at any point it becomes an issue you would subpoena the psychiatrist to get the information you need. Similarly, if there is an active family court action, you can subpoena the records of DHS to *try* to find out who reported alleged abuses.
Does it make any sense that if the claim is valid they would knowingly allow the child to be with the alleged abuser (UNSUPERVISED) half of the time? No it does not, and I hope that tells you what they *likely* believe of the allegations of abuse.
I hope this helps. Please let me know if you have any other questions or you need me to clarify anything. I'd be happy to do so.