Let Y be a principal homogeneous space of an abelian surface, or a K3 surface, over a finitely generated extension of the rational numbers. In 2008, Skorobogatov and Zarhin showed that the Brauer group modulo algebraic classes Br Y/Br_1 Y is finite. We study this quotient for a family of principal homogeneous spaces of abelian surfaces, as well as a related family of geometrically Kummer surfaces of Picard rank 19. Over fields of characteristic 0, we relate the existence of a nontrivial transcendental Brauer class on Y to the commutativity of the Galois group of an associated n-division field. For a fixed number field k, prime l, and a certain geometric N\'eron-Severi lattice, we show that the l-primary part of #(Br Y/Br_1 Y) is bounded, independent of Y. We prove the equivalence among strong uniform boundedness statements for #(Br Y/Br_1 Y), for the existence of abelian n-division fields, and for rational points on certain modular curves.