A private sale DOES NOT REQUIRE A DRIVER LICENSE or ANY OTHER PROOF OF IDENTITY. Do not overthink this!
What the law actually requires in a private sale is the following relevant excerpt from 18 USC § 922(a)(5):
From the OP description, he knows the guy lives here in a home that he owns. There is no worry here. Sell him the gun!: . . . .who the transferor knows or has reasonable cause to believe does not reside in .... the State in which the transferor resides . . .