A birational Torelli theorem with a Brauer class

Let $text{M}_C( 2, mathcal{O}_C) cong mathbb{P}^3$ denote the coarse moduli space of semistable vector bundles of rank $2$ with trivial determinant over a smooth projective curve $C$ of genus $2$ over $mathbb{C}$. Let $beta_C$ denote the natural Brauer class over the stable locus. We prove that if $f^*( beta_{C}) = beta_C$ for some birational map $f$ from $text{M}_C( 2, mathcal{O}_C)$ to $text{M}_{C}( 2, mathcal{O}_{C})$, then the Jacobians of $C$ and of $C$ are isomorphic as abelian varieties. If moreover these Jacobians do not admit real multiplication, then the curves $C$ and $C$ are isomorphic. Similar statements hold for Kummer surfaces in $mathbb{P}^3$ and for quadratic line complexes.