Non-spherical eye tracking - Blender to Unity

So this goes out to those who've successfully used Blender with Unity.

Say I want to make a cartoonish character with eye sockets that won't fit a sphere. Oblong eyes, sphere would grow out the skull from being too large etc. Unity doesn't work with mesh deformers so Lattice deformation is right out the window.

What needs to be accomplished is this: A mesh modeled to fit the eye socket, needs to have a pupil that can be moved to track a position.

I've no idea how this might be accomplished but... What I think should happen: a texture is applied to the mesh and moved around. The mesh itself doesn't move.

This eliminates the issue with a mesh we can't rotate nor deform to rotate. It's down to the texture though. Is there a way to do this?