It would most likely begin as voluntary, to gain acceptance, then later it would become mandatory, and would be unenforceable without registration just like Universal Background Checks.
I can see a lot of people that would not accept UBCs saying this is "reasonable", but it could easily end up causing the same effect.
This, and it could easily become "mandatory" without any further legislation.
"Negligence," as a tort, is the failure to exercise ordinary care, with resultant harm. If NICS (or similar) is readily available to everybody, and people start using it, then the person who doesn't call for a background check would cease to be "ordinary." If he sells a gun without a background check, and the buyer later turns out to use it to commit a crime, a harm occurs. Thus, even without explicit legislation, the courts could impose a de facto mandate to run background checks on private sales by means of tort law.