The background gauge identity at fixed graviton

The pure-gauge part of ${\mathcal{L}}_{\xi }\hat{h}$ is absorbed by the plateau identity; what remains is the linearized Christoffel symbol of $\hat{h}$ contracted with $\xi$. This is not a gauge artefact of the de Donder choice: ${\Gamma }^{(1)}[\hat{h}]$ is defined for every configuration, and the statement is modulo the full ideal.