Yup, that sounds right. 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?