The plane will become lighter.
The birds weight=mg, but they are balancing their weight by the lift they produce by flapping their wings.
Imagine if the birds were outside the fuselage, e.g. on the wings. No one would argue that their weight must be included in the AUW. But if they were to fly off the weight of the aircraft would suddenly decrease. It makes no difference if the birds are inside the fuselage.
TSP
Obviously they would have to have a bloody good grip!