Geometric realizations of Brauer classes on K3 surfaces from hyperkähler contractions

Elements of the Brauer group $\operatorname{Br}(S)$ of a variety $S$ have geometric incarnations as étale-projective $S$-bundles, yet producing minimalist constructions of such bundles, which often power arithmetic applications, remains a difficult problem. When $S$ is a K3 surface with Picard rank