All you need to do to stop that is make it possible for private individuals to conduct or verify a background check without involving an FFL dealer.

Sellers have a responsibility to sell only to non-prohibited people. Without a public background check option, that means you can’t sell if you have reason to believe they are prohibited.

As soon as you provide the option, your refusal to conduct a check stops being exculpatory evidence and starts demonstrating malfeasance.