Consider first the special case where $x_{1}=x_{2}=0,\eta_{p}=\sqrt{-1}y_{p}$. By pairing $(n_{1},n_{2})$ with $(-n_{1},-n_{2})$ in the summation, we obtain

$$\begin{cases}\gamma_{p3}(\eta_{1},\eta_{2},\mu)=\frac{-3a_{p\bar{q}}\sqrt{-1}y_{q}}{8\pi^{2}A^{1/2}}\sum_{(n_{1},n_{2})\in\mathbb{Z}^{2}}\gamma_{+}(n_{1}+\sqrt{-1}y_{1},n_{2}+\sqrt{-1}y_{2},\mu),\\ \gamma_{p4}(\eta_{1},\eta_{2},\mu)=\frac{3a_{p\bar{q}}\sqrt{-1}y_{q}}{8\pi^{2}A^{1/2}}\sum_{(n_{1},n_{2})\in\mathbb{Z}^{2}}\gamma_{-}(n_{1}+\sqrt{-1}y_{1},n_{2}+\sqrt{-1}y_{2},\mu).\end{cases}$$

By the Cauchy integral test,

$$\begin{split}&\sum_{(n_{1},n_{2})\in\mathbb{Z}^{2}}\frac{1}{|(n_{1}+\sqrt{-1}y_{1},n_{2}+\sqrt{-1}y_{2},\mu)|_{a}^{3}a_{p\bar{q}}(n_{p}+\sqrt{-1}y_{p})(n_{q}-\sqrt{-1}y_{q})}\\ \leq&CI_{01}\leq CA^{-1/2}\frac{1}{\varrho^{3}}\max(1,\log(\frac{A\mu^{2}}{|y|_{a}^{2}})).\end{split}$$

Similarly,

$$\begin{split}&|\sum_{n_{1},n_{2}}\frac{1}{(a_{p\bar{q}}(n_{p}+\sqrt{-1}y_{p})(n_{q}-\sqrt{-1}y_{q}))^{2}|(n_{1}+\sqrt{-1}y_{1},n_{2}+\sqrt{-1}y_{2},\mu)|_{a}}|\\ \leq&CI_{02}\leq CA^{-1/2}\frac{1}{|y|_{a}^{2}\varrho}.\end{split}$$

Combining these two estimates,

$$|\gamma_{p3}-\gamma_{p4}|(\eta_{1},\eta_{2},\mu)\leq CA^{-1/2}|a_{pq}y_{q}||\mu|\frac{1}{|y|_{a}^{2}\varrho}\leq\frac{CA^{-1/4}|\mu|}{|y|_{a}\varrho}.$$

Morever, when $\mu\geq 0$,

$$\begin{split}&\frac{1}{a_{p\bar{q}}(n_{p}+\sqrt{-1}y_{p})(n_{q}-\sqrt{-1}y_{q})}|\frac{A^{1/2}\mu}{|(n_{1}+\sqrt{-1}y_{1},n_{2}+\sqrt{-1}y_{2},\mu)|_{a}}-1|\\ \leq&C\frac{1}{|(n_{1}+\sqrt{-1}y_{1},n_{2}+\sqrt{-1}y_{2},\mu)|_{a}^{2}},\end{split}$$

whence

$$\begin{split}&|\sum_{n_{1},n_{2}}\frac{1}{|a_{p\bar{q}}(n_{p}+\sqrt{-1}y_{p})(n_{q}-\sqrt{-1}y_{q})|^{2}}\left(\frac{A^{1/2}\mu}{|(n_{1}+\sqrt{-1}y_{1},n_{2}+\sqrt{-1}y_{2},\mu)|_{a}}-1\right)|\\ \leq&C\sum_{n_{1},n_{2}}\frac{1}{a_{p\bar{q}}(n_{p}+\sqrt{-1}y_{p})(n_{q}-\sqrt{-1}y_{q})|(n_{1}+\sqrt{-1}y_{1},n_{2}+\sqrt{-1}y_{2},\mu)|_{a}^{2}}\\ \leq&C\int_{\mathbb{R}^{2}}\frac{1}{a_{p\bar{q}}(s_{p}+\sqrt{-1}y_{p})(s_{q}-\sqrt{-1}y_{q})|(s_{1}+\sqrt{-1}y_{1},s_{2}+\sqrt{-1}y_{2},\mu)|_{a}^{2}}ds_{1}ds_{2}\\ \leq&CA^{-1/2}\frac{1}{\varrho^{2}}\max(1,\log(\frac{A\mu^{2}}{|y|_{a}^{2}})).\end{split}$$

This leads to

$$|\gamma_{p3}(\eta_{1},\eta_{2},\mu)|\leq\frac{CA^{-1}|a_{p\bar{q}}y_{q}|}{\varrho^{2}}\max(1,\log(\frac{A\mu^{2}}{|y|_{a}^{2}})),\quad\mu\geq 0.$$

Similarly

$$|\gamma_{p4}(\eta_{1},\eta_{2},\mu)|\leq\frac{CA^{-1}|a_{p\bar{q}}y_{q}|}{\varrho^{2}}\max(1,\log(\frac{A\mu^{2}}{|y|_{a}^{2}})),\quad\mu\leq 0.$$

For general $|x_{1}|,|x_{2}|\leq\frac{1}{2}$, the difference $\gamma_{p3}(\eta_{1},\eta_{2},\mu)-\gamma_{p3}(\sqrt{-1}y_{1},\sqrt{-1}y_{2},\mu)$, respectively $\gamma_{p4}(\eta_{1},\eta_{2},\mu)-\gamma_{p4}(\sqrt{-1}y_{1},\sqrt{-1}y_{2},\mu)$, can be estimated by termwise comparing the two series using the methods above. The result is

$$\begin{cases}|\gamma_{p3}(\eta_{1},\eta_{2},\mu)-\gamma_{p3}(\sqrt{-1}y_{1},\sqrt{-1}y_{2},\mu)|\leq\frac{C(|x_{1}|+|x_{2}|)}{A^{1/2}\varrho^{2}}\max(1,\log(\frac{A\mu^{2}}{|y|_{a}^{2}})),\mu\geq 0,\\ |\gamma_{p4}(\eta_{1},\eta_{2},\mu)-\gamma_{p4}(\sqrt{-1}y_{1},\sqrt{-1}y_{2},\mu)|\leq\frac{C(|x_{1}|+|x_{2}|)}{A^{1/2}\varrho^{2}}\max(1,\log(\frac{A\mu^{2}}{|y|_{a}^{2}})),\mu\leq 0,\end{cases}$$

and

$$\begin{split}&|(\gamma_{p3}-\gamma_{p4})(\eta_{1},\eta_{2},\mu)-(\gamma_{p3}-\gamma_{p4})(\sqrt{-1}y_{1},\sqrt{-1}y_{2},\mu)|\leq\frac{C|\mu|(|x_{1}|+|x_{2}|)}{|y|_{a}^{2}\varrho},\end{split}$$

so the claims in the Lemma reduces to the special case $x_{1}=x_{2}=0$ above. ∎