For $f$ with sufficient decay at infinity, we can find $u=P_{g^{(2)}}f$ with estimate $\left\lVert\nabla^{2}u\right\rVert_{C^{k,\alpha}_{\delta,\tau}(\mathbb{C}^{3},\text{ Sym}^{2})}\leq C\left\lVert f\right\rVert_{C^{k,\alpha}_{\delta,\tau}(\mathbb{C}^{3})}.$ Using $\delta+\tau<-1$ and $\delta<-1$, we can integrate from spatial infinity to obtain the required gradient bound.

For a general $f$ without fast decay assumption, take a weakly convergent sequence of fast decaying functions $f_{k}\to f$ bounded in $C^{k,\alpha}_{\delta,\tau}(\mathbb{C}^{3})$, and find $u_{k}=P_{g^{(2)}}f_{k}$ with gradient bounds. After adjusting $u_{k}$ by additive constants to make $u_{k}(0)=0$, we can extract the subsequential limit $u$ of $u_{k}$, which solves $\Delta_{g^{(2)}}u=f$ with the gradient bound. ∎