I am posting as an answer instead of a comment, even though I think this might be wrong, because I could not format the weblinks properly in comments.
Since the (orbifold) moduli space is a quotient of the Siegel upper half space by the (orbifold) fundamental group $\textbf{Sp}_{2g}(\mathbb{Z})$, it seems to me that the analytic Brauer group of the moduli space should be $\text{Hom}(H_2,\mathbb{Q}/\mathbb{Z})$, where $H_2 = H_2(\textbf{Sp}_{2g}(\mathbb{Z}))$ is the kernel of the universal central extension of $\textbf{Sp}_{2g}(\mathbb{Z})$. According to the proof of Proposition 2.3 of
Finite quotients of symplectic groups vs mapping class groups
by Funar and Pitsch, it is well-known that $H_2$ equals $\mathbb{Z}$ for all $g\geq 3$. So this implies that the analtyic Brauer group is $\mathbb{Q}/\mathbb{Z}$.
