But I don't think it's analogous to the bird in the cabin. Surely the bird hanging (getting slightly surreal here) from the cabin roof is not the same as it being in lift-generating flight inside the cabin?
But if it's not, then when it's in flight does it add to the weight of the 'aircraft' as suggested earlier. And if it does, then does that weight go away from the 'aircraft' while the bird is in non-lift-generating descending-flight?
Imagine having a small helicopter, hovercraft and other vehicles in the cargo hold of a big freighter plane - whether the plane is resting on tarmac, in steady taxying or in steady cruise.
When a vehicle is inactive and just sits in the hold, its weight is transferred to the cargo hold through its landing gear.
The weight is actually already carried by air. A vehicle is supported by air trapped in its suspension. And the same vehicle is also supported by air supported inside its tires. (No power is needed to keep it there, because it cannot escape).
Now, put power on the hovercraft. It rises up on air cushion. A hovercraft in steady hover has precisely its entire weight supported by the same cargo floor, because of the extra pressure of air cushion to the floor. The hovercraft does spend some power to hover because the air in air cushion can escape, unlike that inside tires or suspension.
Now, power up your small chopper. It creates downwash. Its weight is no longer supported by its landing gear - but the downwash transfers exact same weight on floor, and perhaps ceiling. So long as it is a steady hover.
If your helicopter is in unsteady flight them the weight of the plane will change. When the copter is in free fall, accelerating downwards, its weight goes away. When the helicopter is accelerating upwards, the weight of airplane will be more than that including the helicopter. Although you should remember to count the acceleration of air inside - the plane does not acquire the weight as soon as the helicopter increases lift, but only when the extra downwash so created hits floor (or ceiling).