First, just to be clear, in general federal law requires that the transfer of a handgun from a resident of one State to a resident of another go through an FFL in the transferee's State of residence.
Whether the way of doing that suggested by the OP will work depends on whether the transfer FFL is agreeable. It would be legal, but some FFLs for their own business reasons want to receive a gun for transfer from another FFL. So the seller and buyer might need to do some looking to find a cooperating FFL.
Also, be advised that the transfer FFL can charge whatever he wants for the service.