= Solution
Write $g_{\mu\nu}=\eta_{\mu\nu}+\epsilon h_{\mu\nu}$. The <linearized inverse metric> is $g^{\mu\nu}=\eta^{\mu\nu}-\epsilon h^{\mu\nu}$, and all quadratic Christoffel products are $O(\epsilon^2)$. The wave-coordinate condition linearizes to
$$
\partial^\mu\bar h_{\mu\nu}=0,
\qquad
\bar h_{\mu\nu}=h_{\mu\nu}-\frac12\eta_{\mu\nu}h.
$$
Using the supplied Ricci formula then gives $G^{(1)}_{\mu\nu}=-\tfrac12\Box\bar h_{\mu\nu}$. Consequently the <Linearized Einstein equations> in <Lorenz gauge in linearized gravity> are
$$
\boxed{\Box\bar h_{\mu\nu}=-16\pi T_{\mu\nu}},
\qquad
\boxed{\partial^\mu\bar h_{\mu\nu}=0}.
$$
This Lorenz condition is the first-order form of the <harmonic coordinate> equations.
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