If there's a question about filing a SAR, we all know the answer -- file.
The filing of a single SAR -- outside of reporting $50M wire transfers to/from the unknown entities -- doesn't truly have much impact with FinCEN. They don't care unless one of the letter agencies has an already open investigation into someone. If this customer is doing everything by the book, they have no concern in the world. Just file it. Nobody is listening.
PS. There is a 99% chance they lied because they were afraid you would close them down and they truly have no other banking options available. If they're a profitable customer and you actually value the relationship, why not reach out and be honest that you know they lied and offer them a chance to explain?