Don't do it.
This guy basically told you that he can't pass a background check.
So he asks you to send the gun in his wife's name to the FFL. Then the wife will go in and fill out the paperwork and the FFL won't know anything fishy is going on. So you, the buyer and his wife have entered into a conspiracy.
Either this is a setup by and anti-group or the ATF or he is a complete idiot.
This guy basically told you that he can't pass a background check.
So he asks you to send the gun in his wife's name to the FFL. Then the wife will go in and fill out the paperwork and the FFL won't know anything fishy is going on. So you, the buyer and his wife have entered into a conspiracy.
Either this is a setup by and anti-group or the ATF or he is a complete idiot.