If you don't have a C&R "03" license then whether or not the firearm is C&R eligible makes no difference to you at all.
A handgun must be transferred to you by a dealer IN YOUR STATE OF RESIDENCE. So yes, the dealer in NC will have to mail the handgun to a dealer in KY, who can then transfer it to you.
(IF, part 1: If you were buying a rifle or shotgun, you could purchase and transfer the weapon in NC, as long as you followed all the laws of both NC and KY.)
(IF, part 2: If you held a C&R license, you could transfer any C&R eligible firearm wherever, handgun or long gun.)