It is my understanding that as long as you have enough information to satisfy 1005.11(b)(1) that your investigation must begin immediately.
The customer's reason for not wanting the card to be hot carded would be something to consider in the investigation, but not something that would delay the investigation.
Also, if the customer has the attitude that they don't care since the bank will cover any unauthorized transactions anyway, then the account may need to be closed before the bank loses more money.
Last edited by NSFW; 06/02/15 06:58 PM. Reason: Added last comment
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It is better to act cautiously beforehand than to suffer afterward.
The answers I give are my opinions. Not legal advice.