So how much does the combination weigh while the bird is descending inside the aircraft with its wings folded? There are then no "air pressure waves", and it is not "part of the aircraft".
Or try this. Suppose the bird is in its seat on the aircraft sitting stationary on the ground. We weigh the combination and get a figure. Then the bird takes to the air inside the aircraft. How much do the scales now register? Does it matter if the cabin has a roof or not? And if we (highly hypothetically) performed the same experiment with the aircraft in flight, now does it matter if there is a roof or not?
Consider it this way: look at a freighter with a crane installed in its ceiling (I think several freighters have them) and a heavy item suspended from that crane.
So long as the item remains hooked at the crane, the weight that works on the landing gear is that of the plane with all its loads - including the item.
Now, a clumsy loadmaster drops the item.
So long as the item is in free fall inside the plane, the weight of the plane decreases and is the weight of the plane and load without the item in free fall.
When the item hits the floor, the weight of the plane increases, and becomes bigger than the weight of the plane and load including the item. This lasts until the item comes to a stop (or breaks through the bottom of the plane). After the item has come to a stop, the weight of the plane is again exactly the weight of the plane and load including the item.