The existence of rational points on the Kummer variety associated to a 2-covering of an abelian variety A over a number field can sometimes be established through the variation of the 2-Selmer group of quadratic twists of A. In the case when the Galois action on A[2] has a large image we prove, under mild additional hypotheses, the Hasse principle for associated Kummer varieties, assuming the finiteness of relevant Shafarevich-Tate groups. This provides further evidence for the conjecture that the Brauer-Manin obstruction controls rational points on K3 surfaces. (Joint work with Yonatan Harpaz)