The nonlinear viscoelastic properties for a series of intercalated nanocomposites of an organically modified montmorillonite and a disordered styrene-isoprene diblock copolymer are reported. The linear-to-nonlinear transition for stress relaxation measurements, the applicability of time-strain separability, and the associated damping function are examined. The nanocomposites exhibit strong shear-thinning behavior thought to result from orientation of the layers in response to the applied shear deformation. We demonstrate that the empirical Cox-Merz rule is inapplicable for such nanocomposites and that steady and dynamic oscillatory flow are not equivalent for such mesostructured materials. The K-BKZ constitutive model is used to predict the steady shear properties from the experimentally measured linear stress relaxation and stress relaxation based damping behavior. While being able to reasonably capture the low shear rate shear stress properties, this empirical model breaks down in trying to model the shear stress behavior at intermediate shear rates and the normal stress behavior at all shear rates and silicate loadings.