This paper introduces two stabilization schemes for the Least Squares Commutator (LSC) preconditioner developed by Elman, Howle, Shadid, Shuttleworth and Tuminaro [SIAM J. Sci. Comput., 27, 2006, pp. 1651--1668] for the incompressible Navier-Stokes equations. This preconditioning methodology is one of several choices that are effective for Navier-Stokes equations, and it has the advantage of being defined from strictly algebraic considerations. It has previously been limited in its applicability to div-stable discretizations of the Navier-Stokes equations. This paper shows how to extend the same methodology to stabilized low-order mixed finite element approximation methods.