That would require changes in the existing laws...The biggest problem is twofold...
Background checks for gun sales are required to go through the FBI system.
Individuals are not allowed to do them...Privacy laws...
Does the third party making the check have to be certified and/or bondecd??? And how would that occur???
The check itself with the FBI results in a simple YES/NO answer...Which makes it easier to do...However the third party would be privy to private information in order to do the check...