As in the proof of Lemma 2.18 we may assume $f$ has compact support to ensure a priori the well definition of $G_{\text{Taub}}f$. We use cutoff functions to decompose $f$ into a sum of functions $f_{n}$ supported on $\{n\lesssim\mu_{2}\lesssim n+1,\text{dist}_{g_{a}}(\cdot,\mathfrak{D}_{1})<C_{2}\}$ centred around points $x_{n}\in\mathfrak{D}$, with Hölder bound $\left\lVert f_{n}\right\rVert_{C^{k,\alpha}(B(x_{n},C_{2}))}\lesssim n^{\tau}$. At a fixed point $x$ bounded away from $\text{supp}(f_{n})$, the contribution $G_{\text{Taub}}f_{n}$ is estimated by $|G_{\text{Taub}}f_{n}|\lesssim n^{\tau}(|x-x_{n}|_{a}+1)^{\epsilon-2}$, where $\epsilon>0$ is any given small number (cf. Corollary 2.17 and notice the translational symmetry of $g_{\text{Taub}}$ along $\mathfrak{D}_{1}$). Elliptic bootstrap gives

$$\left\lVert G_{\text{Taub}}f_{n}\right\rVert_{C^{k+2,\alpha}_{0,0}(B(x,\ell(x)/10))}\lesssim n^{\tau}(|x-x_{n}|_{a}+1)^{\epsilon-2}.$$

Summing over all $n\in\mathbb{N}$,

$$\begin{split}&\left\lVert G_{\text{Taub}}f\right\rVert_{C^{k+2,\alpha}_{0,0}(B(x,\ell(x)/10))}\lesssim\sum n^{\tau}(|x-x_{n}|_{a}+1)^{\epsilon-2}\\ \lesssim&\int_{1}^{\infty}y^{\tau}(\ell(x)^{2}+|\mu_{2}(x)-y|^{2})^{\epsilon/2-1}dy\\ \lesssim&\begin{cases}(|x|_{a}+1)^{\tau}\ell(x)^{\epsilon-1}\quad-1<\tau<1-\epsilon,\\ (|x|_{a}+1)^{\epsilon-2},\quad\tau\leq-1.\end{cases}\end{split}$$

Thus

$$\begin{cases}\left\lVert G_{\text{Taub}}f\right\rVert_{C^{k+2,\alpha}_{-1+\epsilon,\tau}}\leq C,\quad-1<\tau<1-\epsilon,\\ \left\lVert G_{\text{Taub}}f\right\rVert_{C^{k+2,\alpha}_{0,-2+\epsilon}}\leq C,\quad\tau\leq-1.\end{cases}$$

which controls $\left\lVert\nabla_{\text{Taub}}^{2}G_{\text{Taub}}f\right\rVert_{C^{k,\alpha}_{\delta,\tau}}$ under the numerical conditions on weight exponents. ∎