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

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