Conservative answer: How can you know for sure the customer is lying. I have tons of stories where bankers thought there's no way that could happen to only find out later the customer is right. For instance, before people understood skimming, bankers would argue "how could someone use your card if it's always been in your possession?"
Reg E basically says they are innocent until proven guilty. If you prove it was truly authorized, deny the claim. If you can't, honor the claim. This doesn't mean you have to reissue a card to them or that you can't close their account.
Aggressive Answer: Since they are turning the claim over to you, you have the right to investigate. If it's truly a criminal act, ask the customer to come in so you can investigate. Have the police come and take a report - part of your due diligence.
Bank’s may certainly request a customer's cooperation in the investigation and can even suggest a police report be filed but it can't be a condition to starting an investigation or granting provisional credit. Banks can turn the transaction over to the police for investigation and they may require a report to be filed, but the bank cannot tell the customer they have to file a police report to make a claim with the bank.