We assume $C^{-1}\delta_{ij}\leq a_{ij}\leq C\delta_{ij}$ which can be relaxed by scaling. It suffices to solve the complex Monge-Ampère equation

$$(\omega^{(3)}+\sqrt{-1}\partial\bar{\partial}\phi^{\prime})^{3}=\frac{3}{4}\sqrt{-1}\Omega\wedge\overline{\Omega}.$$

In Hein’s analytic package (cf. Section 2.7), the conditions on the ambient metric including $C^{k,\alpha}$ quasi-atlas, existence of a distance-like function with Hessian bounds, and the weighted Sobolev inequality, are robust conditions which are inherited by $\omega^{(3)}$ from $\omega^{(2)}$. The volume form error $E^{(3)}$ has faster than quadratic decay by construction:

$$|E^{(3)}|\leq C(|\vec{\mu}|_{a}+1)^{-4+\epsilon}.$$

Thus Hein’s package provides a potential $\phi^{\prime}$ solving the complex Monge-Ampère equation with decay estimate $|\phi^{\prime}|\leq C(|\vec{\mu}|_{a}+1)^{-2+2\epsilon}$. Elliptic bootstrap gives the bound $\left\lVert\phi^{\prime}\right\rVert_{C^{k+2,\alpha}_{0,-2+2\epsilon}}\leq C,$ so $\left\lVert d\phi^{\prime}\right\rVert_{C^{k+1,\alpha}_{-1,-2+2\epsilon}(\mathbb{C}^{3},\Lambda^{1})}\leq C$, which combined with Lemma 2.25 implies the metric deviation estimate. ∎