Convergence rate of Bates distribution to Normal distribution

Hello,

I am working on an illustration of approximating one distribution with another. Specifically, I am approximating a sample mean of N i.i.d. random variables X_i on a bounded domain [0,1] with a normal distribution N(mean(X),Variance(X)/n). I would like to discuss convergence for random variables X with an arbitrary distribution on a bounded domain [0,1], and I would like to give an illustration based on X being a uniform distribution.

For the former, I can use the Berry-Esseen theorem, giving an upper bound on the maximum difference between the CDF of the normal distribution and the CDF of the sample mean of X.

For the latter, I can show simulations that show a fast convergence. I suspect that I could show that for N<5 or so, it converges faster than the Berry-Esseen theorem bound. I haven't been able to find any general derivation of the convergence rate. I have looked using the names I have seen for the distribution of the mean of n uniform random variables without success: Bates distribution, rectangle mean distribution, uniform mean distribution.

Can anyone point me in the right direction?

Regards,

Eli