= Solution
The <inverse metric> condition is
$$
(\eta^{\alpha\rho}+k^{\alpha\rho})
(\eta_{\rho\beta}+h_{\rho\beta})
=\delta^\alpha{}_\beta.
$$
Keeping only terms linear in the <metric perturbation> gives
$$
k^\alpha{}_\beta+h^\alpha{}_\beta=0.
$$
Thus the <linearized inverse metric> is
$$
\boxed{k^{\alpha\beta}=-h^{\alpha\beta}},
\qquad
\boxed{g^{\alpha\beta}
=\eta^{\alpha\beta}-h^{\alpha\beta}+O(\epsilon^2)},
$$
where indices on $h_{\alpha\beta}$ are raised with the Minkowski metric.
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