This isn't really true. Even if the cargo door is left open, the flying birds exert a reaction force downward on the air. The air then exerts a downward force on the bottom of the airplane. Whether the plane is pressurized or not doesn't really change things. In equilibrium, all the forces sum to zero (Newton's 3rd Law).

For the bird to fly, some force needs to counteract gravity. This is provided by the air. The air transmits this force to the floor of the compartment. Basically, the bird's flying increases the air pressure under it. This pressure acts equally on the floor of the airplane and on the bird (but in opposite directions, summing to zero as required by Newton's 3rd Law for a system in equilibrium). The air pressing down on the floor has the same "weight" as the bird. Even if the door is open.