Even with the drawing I can't quite figure out the inside of the shell. I'd want to use a stepper so I could get 360 rotation. Does that seem feasible?

It's the nature of the beast. With hobby servos you get either 180, or continuous rotation. but, while it won't work correctly in this case, you could also use a 1:2 gear combination to get full 360, or an encoder with a full rotation servo. Or use a stepper motor instead of servo, or a DC motor with encoder.

I am afraid the stepper motor can't be installed inside, but you can check this 360 servo motor http://www.dfrobot.com/inde...