Stop flickering from 2 cameras with different near clipping plane distances

I've run into a problem trying to display and cutaway of a 3D Model of a CT Scan (in this case a head and a brain). I get a flickering when I implement two cameras (one on a head/skull), the other on everything else. The two cameras' rendering during game mode jump back and forth if I change one of their near clipping plane distances (clicking "Set Frame" button).

If I were to zoom in during game play, the menu would start to flicker as well. How can I stop this flickering and just set a near clipping plane that doesn't affect the menu?

I've already tried changing the following values on both cameras: depth; near clipping pane distance; occlusion culling; and adding objects and cameras to different levels.

Main Camera

Second Camera

These two images flicker back and forth ~8 times a second.