Your ability to file or not file a chargeback is governed by Visa/Mastercard operating rules. Without knows the specifics of how the transactions were processed (e.g. via token, 3-D Secure, etc.) we can't possibly say for sure which rule applies here.
Remember that Visa/Mastercard establish liability between the bank and the merchant.
Reg E establishes liability between the bank and the consumer.
Whether you have chargeback rights is not material to your Reg E obligation to investigate and reimburse the customer if you conclude that the transactions were not authorized.
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