Quantum Distortion Gravity (QDG) is a metric-affine theory with gauged Weyl symmetry. We set up its Batalin-Vilkovisky (BV) formulation and combine it with the Wetterich functional renormalization group. The BRST transformations of \( \mathrm{Diff}\ltimes \mathrm{Weyl} \) (with local Lorentz transformations for fermions) are written out, their nilpotency and the classical master equation are established, and the Faddeev-Popov action in unitary gauge is shown to have a field-independent Weyl-ghost determinant. For a regulator that does not respect the BRST symmetry we derive the modified Slavnov-Taylor identity with an explicit regulator term \( \mathcal{K}_k \) and show that the Wetterich flow preserves it; the resulting "third equation" for \( \partial_t \mathcal{K}_k \) is an integrability condition and a diagnostic of truncations, not an independent dynamical law. The 60 non-axial components of the Weyl-covariant distortion are auxiliary with constant propagator, so they form an ultralocal Gaussian block that contributes to no curvature-dependent coefficient; the axial vector is a Proca field and does contribute. The anomaly coefficients of the matter content are \( (c,a)=(11/30,\,53/120) \) for nf=3 Dirac fermions and two Proca fields; the graviton has no such pair, and the anomaly of the gauged Weyl symmetry is open. With the Einstein-Hilbert flow and minimally coupled matter, the QDG-like content has a non-Gaussian fixed point for every nf<10, regulator-robust up to \( n_f\simeq9 \), but at nf=3 it lies at g*=2.73, \( \lambda_*=-0.32 \), outside the range where the truncation is reliable; Weyl-gauged operators, gauge dependence and non-minimal couplings are not included. QDG remains a proposal; the status of each result is stated throughout.