I don't think there's a permanent federal record of single handgun purchases.
There is a permanent record of all guns purchased, long guns and hand guns.
Long story short….
I applied for a Curio & Relics license around the year 2000 or so. There was a problem, my name exactly matches the name of a convicted felon that lives in the next county to me. Big problem. Over the course of trying to straighten this out, I was directed to contact the FBI. Why I don’t know, but it’s what the BATF suggested. While on the phone with the FBI, they were able to tell me every firearm I had purchased, starting with the most recent, and they went back 10 years or so, then the subject changed. As I understand it, no permanent records are supposed to be kept, but trust me, they are keeping such a record.
I never did get that C&R license, it became too much of a problem.