Green Function on $S^2$

Question

I am trying to prove the Green's representation formula on $S^2$. Let $x \in S^2$ be fixed, consider \begin{equation*} G(x,y) = \frac{1}{8\pi}\log[4\sin^4(\frac{r}{2})] \end{equation*} where $r = d(x,y)$ is the distance function on $S^2$. One can readily check that $G$ is singular when $y = x$ and that $\Delta_{S^2,y}G(x,y) = -\frac{1}{4\pi}$ when $y \neq x$ since it satisfies \begin{equation*} G''(r)+ \cot{(r)}G'(r) = -\frac{1}{\text{Vol}(S^2)} \end{equation*} i.e., the Poisson equation written in geodesic radial coordinates. Let $\psi \in C^2(S^2)$, I am trying to prove the representation formula \begin{equation*} \psi(x) = \frac{1}{4\pi}\int_{S^2}\psi(y)dV(y) + \int_{S^2}G(x,y)\Delta_{S^2,y}\psi(y)dV(y) \end{equation*} by splitting the second integral on the right hand side into $$ \underbrace{\int_{S^2 \setminus B_x(\rho)}G(x,y)\Delta_{S^2,y}\psi(y)dV(y)}_{\text{Part I}}+ \underbrace{\int_{B_x(\rho)}G(x,y)\Delta_{S^2,y}\psi(y)dV(y)}_{\text{Part II}}$$, where $B_x(\rho)$ is a geodesic ball with center $x$ and radius $\rho > 0$. I have already showed that $$\text{Part I} \to -\frac{1}{4\pi}\int_{S^2}\psi(y)dV(y)$$ as $\rho \to 0$ and would like to show that $$\text{Part II} \to \psi(x)$$ as $\rho \to 0$. According to this note , one could achieve this by using the Volume Comparison Theorem, but I am not quite sure how that works. Recall that if we are working in a domain in $\mathbb{R}^{n}$, $\text{Part II}$ will actually vanish when $\rho \to 0$. Any insight or familiarity with would be appreciated.