We present a computer-assisted study of "coarse'' stability/ bifurcation calculations for kinetic Monte Carlo simulators using the so-called coarse timestepper approach presented in A. G. Makeev, D. Maroudas, and I. G. Kevrekidis, J. Chem. Phys. 116, 10083 (2002). Our illustrative example is a model of a heterogeneous catalytic surface reaction with repulsive adsorbate- adsorbate interactions and fast diffusion. Through numerical continuation and stability analysis, we construct one- and two-parameter coarse bifurcation diagrams. We also discuss several computational issues that arise in the process, the most important of which is the "lifting'' of coarse, macroscopic initial conditions (moments of adsorbate distributions! to fine, microscopic initial conditions (distributions conditioned on these moments).