Found the answer to my own question.
The BSA requires the following:
(2) If the purchaser does not have a deposit account with the financial institution:
(i)(A) The name and address of the purchaser;
(B) The social security number of the purchaser, or if the purchaser is an alien and does not have a social security number, the alien identification number;
(C) The date of birth of the purchaser;
(D) The date of purchase;
(E) The type(s) of instrument(s) purchased;
(F) The serial number(s) of the instrument(s) purchased; and
(G) The amount in dollars of each of the instrument(s) purchased.
(ii) In addition, the financial institution shall verify the purchaser's name and address by examination of a document which is normally acceptable within the banking community as a means of identification when cashing checks for nondepositors and which contains the name and address of the purchaser, and shall record the specific identifying information (e.g., State of issuance and number of driver's license).
The above requirements clearly indicate that the BSA requires the identity of the individual conducting the transaction, otherwise the requirements would be in terms indicating a TIN would be acceptable and would not ask for a birth date.
My conclusion: "purchaser" = person who comes up to the teller window.
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My opinions are not legal advice and are worth what you paid for them.