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A number of commercial fluids, including synthetic automotive oils, food and consumer products containing polymer additives exhibit weakly rate-thickening responses in the final stages of capillarity-driven thinning, where a large accumulated strain and high extensional strain rate alter the thinning dynamics of the slender liquid filament. Consequently, the capillarity-driven thinning dynamics typically feature two distinct regions at the early and late stages of the filament breakup process, each dominated by distinct mechanisms. These features have been incorporated in a simple Inelastic Rate-Thickening (IRT) model with linear and quadratic contributions to the constitutive stress-strain rate relationship, where the apparent extensional viscosity slowly thickens at high strain rates. We numerically compute the thinning dynamics of the IRT model assuming an axially-slender axisymmetric filament and no fluid inertia. The computational results motivate a new self-similar solution dominated by the second-order stress obtained through a similarity transformation. The new asymptotic solution leads to a self-similar filament shape that is more slender than the Newtonian counterpart and results in a quadratic thinning of the mid-point radius of the filament with time to breakup close to singularity. A new and distinct asymptotic geometric correction factor, $X\approx 0.5778$ is obtained, from which a more accurate true extensional viscosity can be recovered from an interpolated time-varying geometric correction factor based on the magnitudes of different stress components. Finally, we propose a statistics-based protocol to select the best-fit constitutive model using a parameter-free criterion, enabling us to quantify the extensional rheological behavior through capillarity-driven thinning dynamics more systematically on complex rate-thickening viscoelastic fluids.