Rational Chow ring of the universal moduli space of semistable rank two bundles over genus two curves

We determine the rational Chow ring of the universal moduli space of rank $2$ semistable bundles over smooth curves of genus $2$, and show that it is generated by certain tautological classes. In the process, we obtain Chow rings of universal Jacobians over genus $2$ curves with marked Weierstrass points, and the Chow ring of the universal pointed Jacobian. This further provides alternate computations to arxiv:2404.12607 in the genus $2$ case.