Proposition: Fierz–Pauli alone is insufficient

Replacing $S_{\mathrm{EH}}$ by $S_{\mathrm{FP}}$ does not preserve the nonlinear Noether identity even on the plateau.

Proof. Set $\varphi =0$. For the complete Einstein action, ${\mathcal{N}}_{\nu }(e_{\mathrm{EH}})=0$. Its order-one identity requires

Deleting the cubic action deletes $2{\mathcal{H}}_2$ but does not delete $Q_{\nu }(2{\mathcal{H}}_1)$. The latter is not identically zero. For example, where $\gamma =\psi (t)\eta$,

Choose a smooth compact tensor supported in $U$ which has this form near a point where $\dot{\psi }\ddot{\psi }\ne 0$. The proposed quadratic nonlinear identity fails there.