The co-optimization of in-cylinder combustion and after-treatment technology has become a major aspect in engine design and development, with the goal of meeting the increasingly restrictive emission regulations in the transportation industry. Selective Catalytic Reduction is a robust technology to control the emission of NOx, and the injection of urea in water solution is the exhaust tailpipe is a key aspect of its operation. The proposed work uses high-fidelity Computational Fluid Dynamics to characterize the atomization dynamics of the liquid jet in relevant cross-flow conditions. The study focuses on a commercial low-pressure (9 bar) pressure-swirl injector which is characterized in its internal geometry through high-resolution X-ray micro-computational tomography. The internal two-phase flow has been modeled according to the volume-of-fluid approach in a large eddy simulation framework and validated against near-nozzle X-ray radiography measurement. Moreover, characterizing the breakup dynamics for the swirling hollow cone formation, and assessing the influence of the cross-flow in the breakup dynamics was completed. The results have been reported proposing Re-Oh maps and probability density functions of the spray kinematics. Higher cross-flow momentum generates an increase in the jet intact length and a reduction of the liquid droplet diameters. The axial momentum of the jet is affected by the cross-flow already in the near-nozzle region, determining a relevant deviation of the spray velocities. This work aims to inform the initialization of Eulerian-Lagrangian spray models through the assignment of droplet kinematics and static one-way coupling between volume-of-fluid results and Lagrangian spray parcels, to be used for system-size domain simulations.