The coherent interaction between free electrons and optical near-fields enables the active modulation of electron wave packets, which is a mechanism central to photon-induced near-field electron microscopy (PINEM). While existing theories effectively describe these interactions at high kinetic energies, the growing interest in low-energy ultrafast electron microscopy may demand frameworks that explicitly account for finite wave packet geometries and recoil effects. In this paper, we develop a rigorous 3D quantum scattering theory for electron wave packets interacting with time-periodic short-range potentials, with optical near-field modulation serving as the primary motivation. By mapping the time-dependent dynamics into an extended Floquet space, we formally connect the modulation process to time-independent multi-channel scattering. We evaluate the resulting scattering amplitudes using both an exact R-matrix approach and a multi-channel eikonal approximation. In the present 3D formulation, the eikonal approximation recovers PINEM-like probabilities, but now weighted by the incoming wave packet’s transverse profile. For the parameters considered, the comparison between the two methodologies verifies the accuracy of the latter. As a benchmark application, we consider a spherically symmetric oscillating potential. This model demonstrates the generation of distinct energy sidebands and reveals that the modulation strength is sensitive to the transverse focusing of the incident electron pulse.