You guys crack me up - you really do.
I'm going to make a statement: the box is airtight.
So it doesn't matter whether the box has a helicopter, a balloon or a duck in it, nor does it matter whether whatever is in the box is being supported by skids, rotors, heated air or flapping wings.
Whatever mass you put into the box before you sealed it will remain there. The mass of the air will remain unchanged whether it is still or being swirled or heated. Since weight is a function of gravity acting on mass, if the mass stays the same, the weight stays the same.
Actually, I crack myself up - I can't believe I'm actually getting concerned about this! Time for a nice hot cuppa...