I would file it like your first suggestion.
Did the POA sign as a POA? Then he/she did it on the customer's behalf. I don't believe you have to prove who benefited, but to the best of your knowledge the customer did, and the way you completed the CTR indicates that as well.
It's like a signer cashing a check from where they work, payable to cash, and saying they are doing it for the company. You go by what they tell you.