One route to numerically propagating quantum systems is time-dependent density functional theory (TDDFT). The application of TDDFT to a particular system's time evolution is predicated on V-representability, which we have analyzed in a previous publication. Here, we describe a newly developed solver for the scalar time-dependent Kohn-Sham potential. We present and interpret the force-balance equation central to our numerical method, describe details of its implementation, and present illustrative numerical results for one- and two-electron systems in both one-dimensional and three-dimensional grids. Innovations of our numerical implementation include the use of preconditioning when inverting the force-balance matrix and an improved propagation method obtaining the Kohn-Sham potential self-consistently at each step of the propagation. A new characterization of V-representability for one-electron systems is also included, along with possible improvements and future directions.