In earlier work with C.~Monical, we introduced the notion of a K-crystal, with applications to K-theoretic Schubert calculus and the study of Lascoux polynomials. We conjectured that such a K-crystal structure existed on the set of semistandard set-valued tableaux of any fixed rectangular shape. Here, we establish this conjecture by explicitly constructing the K-crystal operators. As a consequence, we establish the first combinatorial formula for Lascoux polynomials $L_{wλ}$ when $λ$ is a multiple of a fundamental weight as the sum over flagged set-valued tableaux. Using this result, we then prove corresponding cases of conjectures of Ross--Yong (2015) and Monical (2016) by constructing bijections with the respective combinatorial objects.