The accurate simulation of the dynamics of polydisperse sprays in unsteady gaseous flows with large-scale vortical structures is both a crucial issue for industrial applications and a challenge for modeling and scientific computing. In a companion paper, we have shown the capability of the Eulerian multi-fluid model to capture the dynamics and evaporation of such sprays and extended it in order to handle finite Stokes number crossing droplet trajectories, by using quadrature method of moments in velocity phase space conditioned on droplet size. Such a study was conducted in the limit of infinite Knudsen numbers. In this paper, we investigate a potential extension of such an approach in order to capture the dynamics of polydisperse spray collisions modeled by a Boltzmann operator at the mesoscopic level of description for finite Knudsen numbers. After deriving the model and presenting the dedicated numerical methods needed to preserve the moment space, we validate this approach and its capability of describing collisional crossing jets by comparing the results for both monodisperse and polydisperse clouds of particles with Lagrangian discrete particle simulations.