(Sketch) Consider $s_{1}=s_{2}=0$. The leading order asymptote of $I_{\Gamma,f}$ is obtained by replacing $f$ with the constant $f(0)$ and replacing $\Gamma$ with $\mathbb{R}^{2}_{s_{1},s_{2}}$. At $s=\sum t_{i}\vec{n}_{i}\in\mathbb{R}^{5}$,

$$\int_{\mathbb{R}^{2}}-\frac{1}{8\pi^{2}|s-s^{\prime}|^{3}}f(0)ds_{1}^{\prime}ds_{2}^{\prime}=-\frac{f(0)}{4\pi R}.$$

We then need to estimate the deviation of $I_{\Gamma,f}$ from this leading asymptote. After writing the surface integral as an integral over $s_{1},s_{2}$ plane, we reduce to the flat graph case $f_{3}=f_{4}=f_{5}=0$. Writing

$$f(s^{\prime})=f(0)+\sum_{i=1}^{2}\frac{\partial f}{\partial s_{i}^{\prime}}(0)s_{i}^{\prime}+O(|s^{\prime}|^{2}),$$

we observe that the linear term does not contribute to $I_{\Gamma,f}(s)$ by parity, and the $O(|s^{\prime}|^{2})$ contribution is bounded by

$$C\int\frac{|s^{\prime}|^{2}}{|s-s^{\prime}|^{3}}ds_{1}^{\prime}ds_{2}^{\prime}\leq C\int\frac{r^{2}}{(r^{2}+R^{2})^{3/2}}rdr\leq C.$$

We now consider the normal first derivative $\frac{\partial I_{\Gamma,f}}{\partial t_{i}}$ for $s_{1}=s_{2}=0$ assuming without loss of generality that $df_{i}(0)=0$. After using the Taylor expansion and parity trick above, modulo bounded terms

$$\begin{split}\frac{\partial I_{\Gamma,f}}{\partial t_{i}}\sim&f(0)\frac{3t_{i}}{8\pi^{2}}\int\frac{1}{(s_{1}^{\prime 2}+s_{2}^{\prime 2}+\sum_{j}(t_{j}-f_{j})^{2})^{5/2}}ds_{1}^{\prime}ds_{2}^{\prime}\\ \sim&f(0)\frac{3t_{i}}{8\pi^{2}}\int\frac{1}{(s_{1}^{\prime 2}+s_{2}^{\prime 2}+R^{2})^{5/2}}(1+\frac{2\sum t_{j}f_{j}}{s_{1}^{\prime 2}+s_{2}^{\prime 2}+R^{2}})ds_{1}^{\prime}ds_{2}^{\prime}\\ \sim&f(0)\frac{t_{i}}{4\pi R^{3}}(1+\frac{1}{5}\sum_{j}t_{j}(\frac{\partial^{2}f_{i}}{\partial s_{1}^{\prime 2}}+\frac{\partial^{2}f_{i}}{\partial s_{2}^{\prime 2}}))\\ =&f(0)\frac{t_{i}}{4\pi R^{3}}(1+\frac{1}{5}(\sum_{j}t_{j}\vec{n}_{j})\cdot\vec{H}),\end{split}$$

where $\vec{H}$ is the mean curvature of $\Gamma$ at the origin.

In the same setup, the tangential first derivative $\frac{\partial I_{\Gamma,f}}{\partial s_{i}}$ is modulo bounded terms

$$\begin{split}\frac{\partial I_{\Gamma,f}}{\partial s_{i}}\sim&\int f\frac{\partial}{\partial s_{i}^{\prime}}\frac{1}{8\pi^{2}|s-s^{\prime}|^{3}}ds_{1}^{\prime}ds_{2}^{\prime}\\ \sim&\int\frac{-1}{8\pi^{2}|s-s^{\prime}|^{3}}\frac{\partial f}{\partial s_{i}^{\prime}}ds_{1}^{\prime}ds_{2}^{\prime}\sim-\frac{1}{4\pi R}\frac{\partial f}{\partial s_{i}^{\prime}}(0).\end{split}$$

The argument for second derivatives are similar. ∎