Remark

The defect cannot be cancelled by letting $\hat{h}$ co-transform: $\hat{h}\mapsto \hat{h}+\kappa {\mathcal{L}}_{\xi }\hat{h}$ is not an automorphism of the free gauge-fixed algebra, since $[P_{\mathrm{gr}},{\mathcal{L}}_{\xi }]\ne 0$ for non-Killing $\xi$, and ${\mathcal{L}}_{\xi }\hat{h}$ is not a solution even when $\hat{h}$ is: $G_{\mathrm{lin}}[{\mathcal{L}}_{\xi }h]=-2G_2[K\xi ,h]$ by covariance of the Einstein tensor at second order. The cure is a second-order quantum metric $\hat{k}$ transforming by ${\mathcal{L}}_{\xi }\hat{h}$, i.e.\ the BRST jet. That is Branch B and is not opened. The mechanism is the one X-B recorded for the scalar pullback, now in the metric slot.