  We investigate obstruction classes of moduli spaces of sheaves on K3
surfaces. We extend previous results by Caldararu, explicitly determining the
obstruction class and its order in the Brauer group. Our main theorem
establishes a short exact sequence relating the Brauer group of the moduli
space to that of the underlying K3 surface. This provides a criterion for when
the moduli space is fine, generalising well-known results for K3 surfaces.
Additionally, we explore applications to Ogg-Shafarevich theory for
Beauville-Mukai systems. Furthermore, we investigate birational equivalences of
Beauville-Mukai systems on elliptic K3 surfaces, presenting a complete
characterisation of such equivalences.
