Let $U_1,U_2,\dots$ be iid rv's uniformly distributed on $[-1,1]$. If a natural number $n$ and real $a_1,\dots,a_n$ vary so that
\begin{equation}
\sum_1^n a_i^2=3\quad \text{and}\quad \max_1^n|a_i|\to0\tag{1}
\end{equation}
(whence $n\to\infty$), then (say) by the Berry--Esseen inequality,
\begin{equation*}
S_n:=\sum_1^n a_i U_i
\end{equation*}
converges to a standard normal rv $Z$ in distribution.

The closeness of the distribution of $S_n$ to normality can be reasonably measured in an infinite variety of ways. One of them, in view of the Esseen smoothing inequality (say; see e.g. Theorem 2.5.2, page 21) is to consider the closeness of the characteristic function (cf) $f_n$ of $S_n$ to the cf $f$ of $Z$ in a neighborhood of $0$. Given (1), we have
\begin{equation}
\ln f_n(t)-\ln f(t)=\sum_1^n\ln\frac{\sin a_i t}{a_i t}-\frac{-t^2}2
\sim-\frac{t^4}{180}\,\sum_1^n a_i^4
\end{equation}
uniformly over all $t$ in any given neighborhood of $0$;
here we use the asymptotic expansion
$$\ln\frac{\sin a t}{a t}-\Big(\frac{-a^2 t^2}6\Big)\sim -\frac{a^4t^4}{180}$$
for $a\to0$ and $t$ in any given neighborhood of $0$.

So, the closeness of the distribution of $S_n$ to normality can be measured by $\sum_1^n a_i^4$, which attains its minimum given the first condition in (1) when the $a_i^2$'s are the same for all $i=1,\dots,n$, that is, when the rv's $a_iU_i$ are identically distributed.