= Solution
Use $A_\beta=\partial_\mu h^\mu{}_\beta$ and $\Box=\eta^{\mu\nu}\partial_\mu\partial_\nu$. The <linearized Ricci tensor and scalar> obey
$$
\delta R=\eta^{\alpha\beta}\delta R_{\alpha\beta}
=\partial_\mu A^\mu-\Box h.
$$
The background <Ricci tensor> and <Ricci scalar> vanish, so varying the metric in the scalar-curvature term contributes nothing at first order beyond this trace. The <Einstein tensor> perturbation is consequently
$$
\boxed{\delta G_{\alpha\beta}=\frac12\left[
-\Box h_{\alpha\beta}+\partial_\alpha A_\beta+\partial_\beta A_\alpha
-\partial_\alpha\partial_\beta h
-\eta_{\alpha\beta}(\partial_\mu A^\mu-\Box h)\right].}
$$
All contractions use the <Minkowski metric>. Direct differentiation gives $\partial^\alpha\delta G_{\alpha\beta}=0$, the linearized contracted <Bianchi identity>, which checks the relative signs. \b[No time derivatives or gauge-dependent terms have been discarded].
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