Some classes of nonlinear complementarity systems, like optimality conditions for generalized Nash equilibrium problems, typically have nonisolated solutions. A reformulation of those systems as a constrained or unconstrained system of equations is often done by means of a nonsmooth complementarity function. Degenerate solutions then lead to points where the reformulated system is nonsmooth. Newton-type methods can have difficulties close to a nonisolated and degenerate solution. For this case, it is known that the LP-Newton method or a constrained Levenberg-Marquardt method may show local superlinear convergence provided that the complementarity function is piecewise linear. These results rely on error bounds for active pieces of the reformulation. We prove that a related result can be obtained for the Fischer-Burmeister complementarity function on the basis of a somewhat different Index Error Bound Condition. To this end, a new convergence framework is developed that allows significantly larger steps. Then, by a sophisticated analysis of the constrained Levenberg-Marquardt method and a corresponding choice of the regularization parameter, local superlinear convergence to a solution with an R-order of 4/3 is shown.