Recent years have seen a growing interest in the thermodynamic cost of dissipative structures formed by active particles. Given the strong finite-size effects of such systems, it is essential to develop efficient numerical approaches that discretize both space and time while preserving the original dynamics and thermodynamics of active particles in the continuum limit. To address this challenge, we propose two thermodynamically consistent kinetic Monte Carlo methods for active lattice gases, both of which correctly reproduce the continuum dynamics. One method follows the conventional Kawasaki dynamics, while the other incorporates an extra state-dependent prefactor in the transition probability to more accurately capture the self-propulsion velocity. We find that the error scales linearly with time step size and that the state-dependent prefactor improves accuracy at high P'{e}clet numbers by a factor of ${Pe}^2$. Our results are supported by rigorous proof of convergence as well as extensive simulations.