The Chow ring of the universal Picard stack over the hyperelliptic locus

Let $\mathscr{J}^d_g \to \mathscr{M}_g$ be the universal Picard stack parametrizing degree $d$ line bundles on genus $g$ curves, and let $\mathscr{J}^d_{2,g}$ be its restriction to locus of hyperelliptic curves $\mathscr{H}_{2,g} \subset \mathscr{M}_g$. We determine the rational Chow ring of $\mathscr{J}^d_{2,g}$ for all $d$ and $g$. In particular, we prove it is generated by restrictions of tautological classes on $\mathscr{J}^d_g$ and we determine all relations among the restrictions of such classes. We also compute the integral Picard group of $\mathscr{J}^d_{2,g}$, completing (and extending to the $\mathrm{PGL}_2$-equivariant case) prior work of Erman and Wood. As a corollary, we prove that $\mathscr{J}^d_{2,g}$ is either a trivial $\mathbb{G}_m$-gerbe over its rigidification, or has Brauer class of order $2$, depending on the parity of $d - g$.