We assess the model discriminating power of a combined phenomenological analysis of and conversion on different target nuclei, including the current hadronic uncertainties. We find that the theoretical uncertainties can be largely reduced by using input from lattice QCD, and do not constitute a limiting factor in discriminating models where one or, at most, two underlying operators (dipole, scalar, vector) provide the dominant source of lepton flavor violation. Our results show that a realistic discrimination among underlying mechanisms requires a measurement of the ratio of conversion rates at the 5% level (two light nuclei) or at the 20% level (one light and one heavy nucleus). We also illustrate these main conclusions in the context of a supersymmetric model.