Recent experiments with heavy ions and planned experiments with ultraintense lasers require nonperturbative solutions to quantum field theory for predicting and interpreting the results. To propel this theoretical direction, we solve the nonperturbative problem of an electron in a strong transverse confining potential using Hamiltonian light-front quantum field theory. We evaluate both the invariant mass spectra and the anomalous magnetic moment of the lowest state for this two-scale system. The weak external field limit of the anomalous magnetic moment agrees with the result of QED perturbation theory within the anticipated accuracy.