Is $\int_{0}^{1}\exp\left(-\frac{n(x-Ks)^2}{2Ks}\right)s^{\alpha-3/2}(1-s)^{\beta-1}\mathop{\mathrm ds}$ approximately Gaussian as $K\rightarrow 0$?

Question

Suppose that $$X\mid p\sim\mathcal{N}\left(Kp,\frac{Kp}{n}\right)$$ for $K^2\approx 0$. Now, if $p\sim\text{Beta}(\alpha,\beta)$ for large $\alpha,\beta$ (say $\alpha\geq1.5$ and $\beta\geq 1$), what is the probability density function for $X_i$? By the law of total probability, this amounts to solving the following integral: \begin{equation} f_{X}(x)\propto \sqrt{\frac{n}{K}}\int_{0}^{1}s^{-1/2}\exp\left(-\frac{n(x-Ks)^2}{2Ks}\right)s^{\alpha-1}(1-s)^{\beta-1}\mathop{\mathrm ds}.\\ \end{equation} I am interested in the case where $K\rightarrow 0$ and, from numerical integration, I find that a normal approximation is perfectly fine (even when $\alpha\not\sim\beta$ and a normal approx. for the Beta distribution is invalid). Why is this?