The short answer is that the closed container would have the same average weight whether the birds were in the air or not. The reason is that the only way to counter the downward force of the birds' weight would be with an equal upward force. In flight that force can only come from an equal downward acceleration force applied to air. (If it is a different mass of air, it would take a different acceleration to match the acceleration of the birds' mass, but that is an arithmetic detail and does not affect the result.)
Actually, even in an open wire container the difference in weight would not be very great; the downdraft of air would accelerate even an open cage significantly, though not as perfectly as might be expected of a closed container.
