OK, now I understand the problem.
And a solution did jump out at me. It is a bit gnarly so buckle in.
- I selected the center triangle for the eye area, then expanded the selection twice with Ctrl+Numpad “+”
- I duplicated that area with Shift+d and separated it with p,s
- I then set the original mesh to make the inner bit transparent - by adding a material with Alpha=0, Specularity=0, and no Shadow - and assigning that to the region defined by that first triangle and only one Ctrl+Numpad “+”
- I then went into the other mesh, the eye chunk, and did the inverse: I made the whole thing the Transparent Material then selected the first triangle and did Ctrl+Numpad “+” one time then assigned that to the old Material.
- It works at this point for render and for the BGE, but it was still looking shabby in the 3D View, so I enabled Object Properties > Display > Transparency for both the main mesh and the eye chunk.
The reason this tomfoolery is required is because of the way smooth shading works. It interpolates the value based on the normals of adjacent faces. If you rip a portion of a mesh out, the edges of the meshes have nothing to interpolate against. So I had to include an invisible edge for both meshes.
Here is the example of what I did: Wendell2.blend (857 KB) (Note that this is only for the Left eye. You could Mirror Modify or similar to get the other side done).
I still say the right way to do this is to use UV map offsets.
Another thing, I notice that your bones are all very short. This is going to mess with your deformations. I see that it already messed with your Automatic Weight assignments. I would select the Wendell Mesh, Alt+p to clear parent and remove the Armature Modifier. Then I would stretch all the bones out so they reached their children, then I would re-parent using Automatic Weights. The weighting will come out 10x better and the pivots for bone deformations will come out 100x better.
What do you think?