Setup: Consider a scalar field $\phi(x)$ which classically satisfies the curved spacetime equation,
\begin{equation} g^{ab}\nabla_a\nabla_b \phi-m^2\phi =0, \end{equation}
where we assume that the background spacetime metric $g_{ab}$ is fixed in the sense that it does not depend on $\phi$ (i.e., the $\phi$-field has negligible contribution to the stress-energy).
Now suppose that the metric can be written as a small perturbation away from Minkowski spacetime,
\begin{equation} g_{ab} = \eta_{ab} + h_{ab}, \end{equation} where the components of $h_{ab}$ are much less than 1 in some global inertial coordinate system of $\eta_{ab}$ (see Wald section 4.4 for discussion). Importantly, further suppose that this perturbation $h_{ab}$ is also fixed (i.e., $\phi$ does not contribute to the stress energy tensor even at the order of $h_{ab}$).
That is, consider a scalar field $\phi(x)$ satisfying the classical equations of motion: [note $g^{ab}\equiv g_{ab}^{-1} = \eta^{ab}-h^{ab}$ so that $g^{-1}_{ac}g_{cb}=\delta^{a}_{\; b}+\mathcal{O}(h_{ab}^2)$]
\begin{align} 0&=g^{ab} \nabla_a\nabla_b \phi-m^2\phi\\ &=(\eta^{ab}-h^{ab})(\partial_a\partial_b-\Gamma^c_{\; ab}\partial_c)\phi-m^2\phi\\ &=(\eta^{ab}-h^{ab})\partial_a\partial_b\phi-\eta^{ab}\Gamma^c_{\; ab}\partial_c\phi-m^2\phi+\mathcal{O}(h_{ab}^2), \label{eom} \tag{*} \end{align}
where $h_{ab}$ is independent of $\phi$ and,
\begin{equation} \Gamma^c_{\; ab}=\frac{1}{2}\eta^{cd}(\partial_a h_{bd} + \partial_b h_{ad} - \partial_d h_{ab})+\mathcal{O}(h_{ab}^2). \end{equation}
Question: If $g_{ab}=\eta_{ab}$, then I might simply introduce global inertial coordinates $(t,\vec{x})$, use these coordinates to Fourier decompose the field into its momentum modes $\phi(x)\propto \int d^3k \hat{\phi}(k)\exp{(\vec{k}\cdot \vec{x})}$, plug this mode expansion into the equations of motion, and then proceed along the lines of Peskin and Schroeder section 2.3 to "quantize" the field. However, it's not clear to me how to proceed along these lines here since the metric is no longer precisely invariant under the Lorentz transformations.
Supposing that the spacetime is globally hyperbolic and static in the asymptotic future/past, then the quantum theory should be well-defined with a prefered vacuum state in the asymptotic future/past. Furthermore, it seems that one should, in principle, be able to obtain the two-point function of the theory, e.g., as some perturbative expansion in orders of $h_{ab}$. Nevertheless, I don't see how to get started in this direction.
