We present a study of the field-dependent dispersion coefficient of point-like particles in various 2D overdamped systems with obstructions (periodic, percolating, and trapping distributions of obstacles). These calculations profit from the synthesis of a newly proposed Monte Carlo algorithm - the first such algorithm that correctly reproduces the free dispersion coefficient in the presence of finite external fields -and an asymptotically exact calculation technique. The resulting method efficiently produces algebraic and numerical results without the need to actually perform Monte Carlo simulations. When compared to such simulations, our exact method features a negligible computational cost and exponentially small errors. Utilizing the power of this numerical method, we engage in comprehensive parametric analysis of several model systems, revealing very subtle effects that would otherwise be swamped by statistical errors or incur prohibitive computational costs. The unified framework presented here serves as a template for further applications of lattice random-walk models of biased diffusion.