Route 3: the held-fixed sign cancels

For a single metric field $q=\eta +\kappa \gamma$, $k=0$ describes the original parametrization, not a prohibition on $\kappa$ dependence after transformation. The genuine flow of $\kappa \xi +{\kappa }^2\zeta$ gives

Its infinitesimal law is $\delta \gamma =K\xi +\kappa {\mathcal{L}}_{\xi }\gamma$. No independent second metric field is needed for this bookkeeping identity.