You're wrong despite your confidence. Cage will have zero net gravitational effect on the black hole (assuming the cage is perfectly symmetrical), but that just

You're wrong despite your confidence. Cage will have zero net gravitational effect on the black hole (assuming the cage is perfectly symmetrical), but that just means it won't by itself exert gravitational force on the black hole (and vice versa) causing it to move in any direction. Cage's material will still get attracted to the black hole and close to the event horizon structural forces exerted by gravity will be hard / impossible to counteract with current materials: https://en.wikipedia.org/wiki/Strength_of_materials https://en.wikipedia.org/wiki/Strength_of_materials