I’d be okay with an optional mechanism to do a background check where a private seller could call in your driver’s license number and no other information (different from SSN in my state) and immediately hear whether you were permitted to own a firearm. That strikes me as useful. But a background check that includes a description of the firearm, including the model and serial number, is clearly gun registration, and that can have no rational purpose other than to infringe on my right to keep and bear arms.
Do sellers of knives do background checks?
Baseball bats?
Cans of gasoline?
Cars?