As to your question about an agreement with Bank A, if the customer is a consumer, Regulation E's protections trump any agreement. If the consumer files an error resolution claim, or a claim under 205.6, the bank has to adhere to Regulation E, its contract notwithstanding.
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John S. Burnett
BankersOnline.com
Fighting for Compliance since 1976
Bankers' Threads User #8