We implement a computer-assisted approach that, under appropriate conditions, allows the bifurcation analysis of the "coarse" dynamic behavior of microscopic simulators without requiring the explicit derivation of closed macroscopic equations for this behavior. The approach is inspired by the so-called time-stepper based numerical bifurcation theory. We illustrate the approach through the computation of both stable and unstable coarsely invariant states for kinetic Monte Carlo models of three simple surface reaction schemes. We quantify the linearized stability of these coarsely invariant states, perform pseudoarclength continuation, detect coarse limit point and coarse Hopf bifurcations, and construct two-parameter bifurcation diagrams.