Form 4473 remains in the FFL dealer files. Background check (phone or on-line) is not allowed to be retained under federal law. A national registry of firearms and owners is not allowed under current federal laws.
With automated notification of each transaction, and the credit card data will include cardholder identification linked to credit bureau reports providing date of birth, employment history, residence history, etc. With an ever-growing list of firearms and ammo purchasers it would be pretty easy to establish a target list for taxation or confiscations.
We are already seeing federal officers going to peoples' homes inquiring about lawful transactions (multiple sales, etc). How pleasant would life be with LEOs tramping around your neighborhood, making inquiries of your neighbors, maybe stopping by your place of employment and continuing their "investigation" by hounding your supervisor, HR director, co-workers, etc?
The 4th Amendment guaranteeing privacy in one's person, house, papers, effects, communications, etc, would seem to require court orders based upon probable cause to believe that a crime has been committed. Should that not apply to my private contracts and transactions with a credit card company?
Investigative fishing trips, back door method for gathering data that is not permitted to be gathered or kept via firearms sales data or background checks, with strong potential for large scale harassment under color of legal authority.