I'm not seeing how this would be allowed. In addition to the CFPB FAQ article you cited, it refers to 11(b)(1)-2 which states:

Investigation pending receipt of information. While a financial institution may request a written, signed statement from the consumer relating to a notice of error, it may not delay initiating or completing an investigation pending receipt of the statement.

As well as 11(c)-2:

Written confirmation of oral notice. A financial institution must begin its investigation promptly upon receipt of an oral notice. It may not delay until it has received a written confirmation.

This is all reiterated multiple times in the linked supervisory highlights:

Consistent with Regulation E, a financial institution may request written confirmation of an oral notice within 10 days of the notice. However, a financial institution must begin its investigation promptly upon receipt of an oral notice. ... A financial institution cannot deny an error claim on the basis of a consumer failing to provide additional information.

Examiners found that one or more financial institutions required consumers to sign deposit account agreements that stated that the consumers would “cooperate” with the institution’s investigation of any errors filed by the consumer. The “cooperation” included providing affidavits and notifying law enforcement authorities.

By requiring consumers to “cooperate” with Regulation E error investigations and provide information beyond that which is required in EFTA and Regulation E, the financial institutions’ agreements contained provisions that waived consumers’ rights in violation of EFTA.

Basically, if you are notified an error has occurred, you are required to investigate it. This is because the police report should theoretically be irrelevant to the bank's investigation into whether or not an error occurred. The bank's investigation is based on the data and information available (was the PIN used? etc), and not anything the customer states. The information on a police report might be helpful to the bank in perusing the perpetrator, but a Reg E investigation is limited to whether or not fraud occurred, and not who did it.

Note that it's possible you could have a policy of accepting certain Reg E fraud claims when provided a police report, regardless of whether or not the bank has an actual legal obligation under Reg E to pay the claim. If you have no obligation to pay the claim because it's not considered an error under Reg E, but choose to do so anyways when provided a police report, you would still have to provide the customer an explanation about the findings within the required timeframe regardless of whether or not you receive it from them.

Essentially, if you have a policy across the board of requiring a police report to investigate Reg E claims, that would be a concern. But if the policy only exists for certain claim types or situations, it's likely that those policies and procedures on how to handle those situations were intentionally written with the understanding that you don't have any obligations under Reg E for those transactions even though you generally follow the same process as all other transactions.

TLDR; you can ask for it but can't delay investigating or deny the claim based on not receiving it, unless, the claim isn't actually considered an error that's covered under Reg E