Say $X_1,\ldots,X_n$ are iid, bounded, symmetric random variables with mean $0$, variance 1 and a smooth density. At what scale does $\frac{X_1+\cdots+X_n}{\sqrt{n}}$ "look like" the normal distribution?
If the $X_i$ are uniform random variables on $[-1,1]$ then it seems that a modification of the characteristic function proof of Berry-Esseen gives that $$\mathbb{E}f\left(\frac{X_1+\cdots+X_n}{\sqrt{n/3}}\right) = \mathbb{E}f(G)+ O(n^{-1} \||f\||_1) + O(e^{-cn} \||D^2f\||_2)\,,$$ where $G$ is the standard unit Gaussian on $\mathbb{R}$. This suggest that $\frac{X_1+\cdots+X_n}{\sqrt{n}}$ "looks" gaussian down to exponential scales, however I am unable to find a reference addressing this situation. The standard examples showing sharpness of Berry-Esseen usually involve discrete random variables. But this calculation suggests that an improved Berry-Esseen holds on much smaller scales when the distribution of $X_i$ is smooth.
Thanks in advance.