Self-regularized entropy: What does black hole entropy predict for tests of Kerr no-hair theorem?

We compute the canonical (brick-wall) entropy of Hawking radiation in a in a quantum black hole model whose strong-field exterior is modeled phenomenologically, to first order in a small quadrupole parameter, by the static q-metric, which is an exact vacuum solution of the Einstein equations. WKB counting of trapped near-horizon cavity modes shows that, within the perturbative small-deformation regime studied here, a modest quadrupolar deformation self-regularizes the ultraviolet divergence: the entropy becomes finite without an ad hoc cutoff. Adopting the Hawking temperature and the Bekenstein-Hawking entropy of a Schwarzschild black hole of the same mass as external thermodynamic inputs, matching this canonical entropy to that benchmark yields an entropy-motivated deformation scale which, when interpreted phenomenologically in a stationary extension, corresponds to percent-to-tens-of-percent violations of the Kerr multipole relations, and provides concrete observational targets for the Next Generation Event Horizon Telescope (ngEHT), the Laser Interferometer Space Antenna (LISA), and planned third-generation (3G) ground-based gravitational wave observatories.