Why does FDTD and FIT disregard Gauss's law?

This is a reformulation of a question I asked a couple of days ago. I’m posting it again because I believe the previous post was very unclear, I will probably delete the previous question.

My question is, why does the FDTD and FIT (Finite integration technique) method for solving Maxwell’s equations disregard Gauss’s Law?

As I understand it, the update equation for FDTD is derived by replacing all the derivatives on Maxwell’s equations for discrete differences on a Yee grid and solves for the next field values, but the update equation is derived only from Ampere’s and Faradays law.

A similar thing happens with FIT, each Maxwell’s equation is discretized, and then only Ampere’s law and Faraday’s law are used to solve for the update equation.

My goal is to build a visual simulation of a moving charged particle with my own code and visualize the electric and magnetic field around the particle. But this independence of Gauss’s law doesn’t allow me to program in an electric charge. Is there a way around this, or is this a limitation of FDTD and FIT? Are there methods that allow the simulation of this scenario or is it unrealistic to do so?