  Suppose that a Hilbert scheme of points on a K3 surface S of Picard rank one
admits a rational Lagrangian fibration. We show that if the degree of the
surface is sufficiently large compared to the number of points, then the
Hilbert scheme is the unique hyperk\"ahler manifold in its birational class. In
particular, the Hilbert scheme is a Lagrangian fibration itself, which we
realize as coming from a (twisted) Beauville-Mukai system on a Fourier-Mukai
partner of S. We also show that when the degree of the surface is small our
method can be used to find all birational models of the Hilbert scheme.
