The framework of "time-dependent basis light-front quantization" (tBLFQ) offers a nonperturbative approach to scattering problems in external fields, based on Fock space truncation. Here we extend tBLFQ to include spatio-temporal field inhomogeneities in multiple spacetime directions. This extension is necessary for the proper modeling of e.g. intense laser fields. We focus on the example of nonlinear Compton scattering of an electron on an axicon-type laser, with an emphasis on the transverse structure of the beam. We analyze the impact of field intensity and particle energy, as well as basis truncation effects, on the radiation spectrum of the scattered electron.