  Conditional on finiteness of relevant Shafarevich--Tate groups, Harpaz and
Skorobogatov used Swinnerton-Dyer's descent-fibration method to establish the
Hasse principle for Kummer varieties associated to a 2-covering of a
principally polarised abelian variety under certain largeness assumptions on
its mod 2 Galois image. Their method breaks down however when the Galois image
is maximal, due to the possible failure of the Shafarevich--Tate group of
quadratic twists of A to have square order. In this work we overcome this
obstruction by combining second descent ideas in the spirit of Harpaz and Smith
with results on the parity of 2-infinity Selmer ranks in quadratic twist
families. This allows Swinnerton-Dyer's method to be successfully applied to K3
surfaces arising as quotients of 2-coverings of Jacobians of genus 2 curves
with no rational Weierstrass points.
