An implementation of the strong-permittivity-fluctuation theory (SPFT) is presented in order to estimate the constitutive parameters of a homogenized composite material (HCM) which is both cubically nonlinear and anisotropic. Unlike conventional approaches to homogenization, the particles which comprise the component material phases are herein assumed to be small but not vanishingly small. The influence of particle size on the estimates of the HCM constitutive parameters is illustrated by means of a representative numerical example. It is observed that, by taking the nonzero particle size into consideration, attenuation is predicted and nonlinearity enhancement is somewhat diminished. In these respects, the effect of particle size is similar to that of correlation length within the bilocally-approximated SPFT.