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I've run a large Thule box on mine and yes, the factory bars will flex, but it should still be plenty stable to keep your box in check. My hypothesis is that the nose of your box is right at the point where the airflow will keep splitting above or below it as the bars flex, increasing the amplitude of the flex as it goes. My advice would be to play around with moving the box forward and back - either by sliding the box on its mounts, or by sliding the bars themselves backward and forward, whichever is easier, to find a point where it doesn't do that any more.
Did you find by moving it more rearward that the bounce was reduced?
Mine was fairly long, so moving it back any further would've caused interference with the liftgate. I had to move mine forward so that the nose overhung the sunroof by a little bit:
It will make it more solid, but still won't resolve the aerodynamic effect that's causing it. It's known as aeroelastic flutter:
