We study the weakly hyperbolic system of conservation laws which arises when we take velocity moments of a Williams-Boltzmann equation studied in gas-particle flows. Such approaches naturally degenerate toward the pressureless gas system of equation in the context of monokinetic velocity distributions (Massot et al. 2009; Kah 2010; Runborg 2000). Quadrature-based numerical algorithms have been proposed in Jin & Li (2003); Gosse et al. (2003) and Desjardin et al. (2008) independently, from (Bouchut et al. 2003), using a first order kinetic-based finite volume method. The computation of the cell-centered fluxes by means of the quadrature abscissas and weights ensures realizability and singularity treatment. Such a quadrature approach and the related numerical methods have been shown to be able to capture particle trajectory crossing (PTC) in a direct numerical simulation (DNS) context, where the distribution in the exact kinetic equation remains at all times in the form of a sum of Dirac delta functions. This paper introduces a fully second-order in time and space transport scheme for this quadrature-based closure, with linear reconstructions for both weights and abscissas. Whereas realizability is ensured, we suggest an algorithm in order to ensure both conditions: maximum principle for velocity and moment vector conservation, in one or two-dimensional configurations.