Limits of Symmetry in Schwarzschild: CKVs and BRST Triviality in the Kerr-Schild Double Copy

We complete our investigation into the residual symmetries of the Kerr-Schild double copy for the Schwarzschild solution. In the first paper in this series, we showed that the infinite-dimensional residual gauge algebra collapses to the finite-dimensional global isometries when restricted to Killing vectors. Here, we extend the analysis to proper conformal Killing vectors (CKVs), solving the field equations via the method of characteristics to obtain explicit conformal solutions. While asymptotic flatness and horizon regularity remove divergent contributions, the surviving components form a non-trivial infinite-dimensional algebra, revealing a structural mismatch with the canonical Schwarzschild solution. We resolve this by constructing a unified, Weyl-compensated BRST complex, showing that the infinite-dimensional modes are BRST-exact and do not correspond to physical degrees of freedom. This demonstrates the quantum consistency of the Kerr-Schild double copy, confirming that the physical spectrum is restricted to global isometries.