This solution will be vastly inferior for catching potential cheaters than having them take a test in the classroom. Requiring an invigilator to inspect individual camera feeds isn't feasible. Furthermore, as the student has full control of the environment, he'll be able to easily work around the controls. For instance, although there is a 360 degree webcam, I suppose it is not monitoring the vertical plane and someone could put answers below or above the camera.
Isn't it more fundamental than that? You're giving someone with huge incentives to cheat access to the hardware. How can you possibly make that system secure?