We examine the linear stability of the one dimensional inhomogeneous (shear banded) pressure-driven flow through a rectilinear microchannel predicted by the VCM model (Vasquez et al., A network scission model for wormlike micellar solutions I: model formulation and homogeneous flow predictions. J.. Non-Newton. Fluid Mech., 144:122-139, 2007). The VCM model is a microstructural network model that incorporates the breakage and reforming of two elastically-active species (a long species 'A' and a shorter species 'B'). The model consists of a set of coupled nonlinear partial differential equations describing the two micellar species, which relax due to reptative and Rousian stress-relaxation mechanisms as well as breakage events. The model includes nonlocal effects arising from stress-microstructure diffusion and we investigate the effect of these nonlocal terms on the linear stability of the pressure-driven flow. Calculation of the full eigenspectrum shows that the mode of instability is a sinuous (odd) interfacial mode, in agreement with previous calculations for the shear-banded Johnson-Segalman model (Fielding and Wilson, Shear banding and interfacial instability in planar Poiseuille flow, J. Non-Newton. Fluid Mech., 165:196-202, 2010). Increased diffusion, or smaller characteristic channel dimensions, smoothes the kink in the velocity profile that develops at the shear band and progressively reduces the spectrum of unstable modes. For sufficiently large diffusion this smoothing effect eliminates the instability entirely and restabilizes the base (shear-banded) velocity profile. (C) 2011 Elsevier B.V. All rights reserved.