Fractional Chern insulators are interaction-driven topological states that realize the lattice analogue of the fractional quantum Hall effect without external magnetic fields. Here we propose a programmable platform for engineering topological flat bands by combining a Zeeman field with a tunable electrostatic superlattice potential in a Rashba spin-orbit-coupled thin film. The Zeeman field generates a nearly flat segment of the Rashba band, which is isolated by the superlattice potential into a flat band with Chern number C=1. Using many-body exact diagonalization, we show that this band supports fractional Chern insulators at filling factors n = 1/3 and 2/3, both of which remain robust over broad parameter ranges. We further identify realistic material candidates and the corresponding device conditions that enable experimental realization. These results establish a versatile and experimentally accessible platform for engineering topological flat bands and exploring correlated topological phases.