For a 360 degree product shot , I would set up lights and camera , and then rotate the product rather than animate the camera.  THis works fine for things like cars or clothes.

If it needs to be a real flyaround then you either need to provide a fairly even 360 light which is not that hard (3 spotlights even placed ) or parent the lights to a ball prop and rotate the ball as you move your camera. (position the ball prop out of camera view above your scene)