Another way of thinking about it would be to imagine a big tub of water on a scale. Suppose the tub is full to the brim. If we drop in a frog that starts swimming around in the tub, does the weight registered on the scale increase?

If the tub is full to the brim, the water displaced by the frog will slosh over the side of the tub. This will exactly match the weight of the frog, so the weight registered by the scale will not change at all. (weight of full tub of water + weight of frog - weight of water displaced by frog = weight of full tub of water).

On the other hand, if the tub is not full, so the water displaced by the frog stays in the tub, the weight registered by the scale will increase by the weight of the frog.

But this implies that the plane-and-bird system above will weigh P (weight of plane) + B (weight of bird, or fly), even if the bird is in flight inside the plane. Thinking about the frog in the tub, suppose the tub full of water weighs T. We add the frog, which weighs F. The tub with the frog swimming it it weighs T + F. If the frog swims over to the side of the tub and climbs up on the edge, the system still weighs T + F. The weight of the system doesn't change depending on whether the frog is swimming or not, so the weight of the plane and bird system shouldn't change just because the bird is flying.