The error $\Delta_{g^{(2)}}(\chi_{1}G_{\text{Taub}}f)-f$ comes from two sources: the deviation of the metric $g_{\text{Taub}}$ from $g^{(2)}$, and the cutoff error. The metric deviation error is estimated in Lemma 2.6 which we recall as $\left\lVert g_{\text{Taub}}-g^{(2)}\right\rVert_{C^{k,\alpha}_{0,-1}}\leq C.$ In particular for $|\vec{\mu}|_{a}>C_{3}$ and on the support of $\chi_{1}$, we have $\left\lVert g_{\text{Taub}}-g^{(2)}\right\rVert_{C^{k,\alpha}_{0,0}}\leq CC_{3}^{-1},$ so the metric deviation error is $O(C_{3}^{-1})$.

We turn to the cutoff error. By Lemma 2.20

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

which implies

$$\begin{cases}\left\lVert\nabla^{2}_{\text{Taub}}(\chi_{1}G_{\text{Taub}}f)\right\rVert_{C^{k+2,\alpha}_{-3+\epsilon,\tau}}\leq C,\quad-1<\tau<1-\epsilon,\\ \left\lVert\nabla^{2}_{\text{Taub}}(\chi_{1}G_{\text{Taub}}f)\right\rVert_{C^{k+2,\alpha}_{-2,-2+\epsilon}}\leq C,\quad\tau\leq-1.\end{cases}$$

so in particular on $\text{supp}(d\chi_{1})\cap\{|\vec{\mu}|_{a}>2C_{3}\}$ where $\ell\sim C_{3}C_{1}^{-1}$, we have

$$\left\lVert\nabla^{2}_{\text{Taub}}(\chi_{1}G_{\text{Taub}}f)\right\rVert_{C^{k,\alpha}_{\delta,\tau}}\leq CC_{3}^{-\epsilon}.$$

By the support assumptions $f=0$ on $\text{supp}(d\chi_{1})\cap\{|\vec{\mu}|_{a}>2C_{3}\}$, hence the cutoff error is $O(C_{3}^{-\epsilon})$. Combining the two errors give the claim. ∎