[scythe]

That's a clever argument, though I wonder if it would be defeated by some restrictions on the chromatic polynomial. In your example, the number of 3-colorings up to isomorphism is extremely large, because the colors of any brick and its cobrick can be switched, and the mortar condition implies the districts are 3-colorable.

Of course, the chromatic polynomial is #P-complete, so this may pose some difficulty.