Actually, yes it does make it weight more, while the plane is still in the air, but over the carrier:
While the force the plane actually exerts downwards on the carrier does not directly increase the carrier's weight, that same force does lower the carrier (minutely) in the water as it displaces slightly more water.
Therefore, since the carrier is now less far from the center of the planet, the force excerpted by gravity (theon the mass of the carrier) is now (minutely) stronger, so the carrier weighs more.