First the Bank needs to stop using "Reg E" and "Merchant "dispute" as antonyms. I can't put graphics in an HTML based post, but if I could, I would use a Ven diagram.
Reg E breaks claims in to "Reg E Electronic Funds transfer errors as defined in 1005.11(a)" and "Non-Reg E errors"
Visa breaks claims into "Fraud" and "Merchant Disputes".
Fraud and Electronic Funds Transfer Errors would both be inside the overlapping part of VISA Fraud claims and Reg E Errors. However, there are some claims that VISA defines as a "merchant dispute" that are also electronic funds transfer errors under 1005.11(a).
For example, if I am charged the wrong amount, Reg E considers that an incorrect EFT and Visa considers it a merchant dispute.
If I cancel a recurring charge, Reg E considers any changes after cancellation as an unauthorized EFT because I revoked the authorization. Visa considers that a merchant dispute.
The Visa chargeback process is one method of investigating Reg E error claims. However, Reg E doesn't care whether you have chargeback rights or whether you can recover funds from the merchant via the chargeback process. Reg E decides who is liable for an error between the bank and the customer. Visa decides who is liable between the bank and the merchant. You cannot deny a Reg E claim because the customer doesn't jump through Visa chargeback hoops. If you don't have what you need to file a chargeback, you either reimburse the customer or contact the merchant directly to try and obtain more information about the authorization to see if you can obtain information that would lead you to conclude that no error occurred.
Have you looked at the CFPB REG E FAQs?
Unauthorized FAQ #9 makes it clear that you cannot require the cardholder contact the merchant as a condition of investigating a claim.