We study the nonlinear optical response of a fishnet structure-metamaterial all-optical switching device that exhibits two near-infrared negative-index resonances. We study and compare the nonlinear optical response at both resonances and identify transient spectral features associated with the negative index resonance. We see a significantly stronger response at the longer wavelength resonance, but identical temporal dynamics at both resonances, providing insight into separately engineering the switching time and switching ratio of such a fishnet structure metamaterial all-optical switch. We also numerically reproduce the nonlinear behavior of our device using the Drude conductivity model and a finite integration technique over wide spectral and pump fluence ranges. Thereby, we show that beyond the linear properties of the device, the magnitude of the pump-probe response is completely described by only two material parameters. These results provide insight into engineering various aspects of the nonlinear response of fishnet structure metamaterials.