There seems to be some confusion on this thread between 'weight' and 'mass'.
Weight is the force of attraction of the earth on a given mass. For one to say something weighs 'less' on the moon, one is modifying the definition to produce a 'moon weight': the force of attraction of the moon on a given mass.
The force of attraction is gravity, and the gravitational force between two bodies is proportional to their masses and inversely proportional to the square of the distance between them. The geese exert a gravitational ‘pull’ on the earth, as does the earth on the geese. It just happens the latter is somewhat stronger!! It is not possible to talk about the ‘weight of the earth’, because it is not possible to have a weight relative to the same body that is producing the gravitational pull. The earth has mass.
The mass of a body is the constant of proportionality between the force applied to a body and the acceleration produced proportional to the force. (Ignoring the concept of 'rest mass' as mass does actually vary with velocity ref: the theory of relativity). A body with mass of 1kg will have this mass regardless of location - the surface of the earth, the surface of the moon, or outer space.
So, for an object to have 'weight', we need gravity and mass. Mass is a constant and weight varies with gravity. The earth’s gravity decreases the further we proceed away from it.
If we load an aircraft very precisely to MAUW at sea level, and then very precisely do the same at an airfield at 5000’, we will find we can fit more in ‘by weight’. If you feel like proving this, just weigh something on an accurate balance at sea level, and try the same at the top of Mount Everest - but wrap up warm! However, in aviation operations the difference is immaterial, and everyone feels more comfortable with the word ‘weight’ - they feel they know what it means! (If anyone knows different - eg calculations are done with large aeroplanes to make altitude adjustments to take on more cargo or fuel at, for example, Kathmandu, I would be fascinated to know!)
Turning to our geese.
The total mass of the aeroplane and cargo remains constant when the geese attempt to start their migration. If there was no variation in gravity, then the weight of the aeroplane would stay constant (closed system, downward force of flapping wings, Newton’s third law etc).
However, the poser of the question in Pilot was dealing with ‘precision’ - after all he was talking about very small centre of gravity movements. In his answer he made the classic mistake of talking about the ‘weight’ of the aircraft not altering by ‘one iota’. If one is being precise, then the weight of the aircraft does indeed decrease. The geese become further from the centre of the earth; they therefore weigh less; the downward force to keep them airborne becomes less; the reduction in the downward force on the floor of the aircraft means that the total weight of the aircraft is less - albeit by a very tiny, tiny amount. If the total weight of the fuselage is less, then the wing loading will decrease.
Unfortunately, the altitude of the aircraft is not given. If it was, we could calculate the reduction in weight of the aeroplane when the geese fly around. It would be a very small figure!
Phew!! Got that off my chest. I must have too much time on my hands!! Actually, kicking my heels waiting for a work permit to come through!