Why do Republicans participate in these rigged “debates?”
Because the debates allow expression of good ideas to a wide audience, albeit under hostile circumstances. A better question might be, "Why don't liberal Democrats participate in a debate with Rush Limbaugh as the moderator?"
Republicans have met liberal Democrats on their own rigged turf and still wipe the floor with them when it comes to ideas that are best for a people who love freedom, smaller government, truth, and general moral rectitude.