CLT for the gap between two Beta(1,2) sums

Take $648$ independent $\text{Beta}(1,2)$ random variables, split into $X_1,\dots,X_{324}$ and $Y_1,\dots,Y_{324}$. Put $S=\sum_{i=1}^{324} X_i$ and $T=\sum_{i=1}^{324} Y_i$. Using a normal approximation, $P(S-T>10)\approx\Phi(a)$, where $\Phi$ is the standard normal CDF. What is $a$?

$a=-5/3$. A $\text{Beta}(1,2)$ variable has mean $\tfrac13$ and variance $\tfrac1{18}$. So $S-T$ has mean $0$ and variance $2\cdot324/18=36$, i.e. standard deviation $6$, and $P(S-T>10)\approx 1-\Phi(10/6)=\Phi(-5/3)$.