Tom, the spring device will have a platform on which to place the object measured. The platform will have a top and a bottom surface. If the air flows freely, the force acting upwards on the bottom will be exactly equal to the force acting downwards at the top. Increase in atmospheric pressure will affect both equally, and will cancel out. If of course the air does not flow freely, and the pressure at the bottom is different from the top, this will affect the apparent weight. This will not normally happen.