No, he does not have to have a copy of your dealer's FFL. In fact, dealers are not obligated to provide a copy of their license to non-licensees when performing an interstate transfer. This is to prevent non-licensees from posing as licensees and illegally ordering firearms from a manufacturer or distributer. A valid license number that he can verify with FFL Easy check is legally sufficient.
That said, the seller would in my opinion be within his rights to insist on a paper copy of the FFL that the firearm is going to. It's his gun and he doesn't have to sell it if he doesn't want to. Some people feel more secure having a paper copy of the license in their files, in case the BATFE shows up on their door step saying that they traced a gun from a crack house to them.
In the past, I have been able to work it out by having a local FFL ship the gun to the buyer's FFL. This way my FFL can get a copy of the receiving FFL's license and I have documentation (on the dealer's bound book) that the gun went out from there. It's a win/win, and dealers usually feel more comfortable receiving guns from other dealers anyways.
Interstate transfers are a fucking pain in the dick. There is alot at stake and no room for error.