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

Let Jᵈg Mg be the universal Picard stack parametrizing degree d line bundles on genus g curves, and let Jᵈ₂, ₆ be its restriction to locus of hyperelliptic curves H₂, ₆ Mg. We determine the rational Chow ring of Jᵈ₂, ₆ for all d and g. In particular, we prove it is generated by restrictions of tautological classes on Jᵈg and we determine all relations among the restrictions of such classes. We also compute the integral Picard group of Jᵈ₂, ₆, completing (and extending to the PGL₂-equivariant case) prior work of Erman and Wood. As a corollary, we prove that Jᵈ₂, ₆ is either a trivial Gₘ-gerbe over its rigidification, or has Brauer class of order 2, depending on the parity of d - g.