Singly-charged anomalous cosmic rays (ACRs) can give rise to multiply-charged ACR ions when they suffer further ionization during acceleration at or near the solar-wind termination shock. Measurements at 1 AU by SAMPEX have shown that above ≈ 25 MeV/nucleon ACR nitrogen, oxygen, and neon ions are multiply charged. These observations have also established that the transition from mostly singly-charged to mostly multiply-charged ACRs occurs at a total kinetic energy of ≈ 350 MeV. Recent simulations for ACR oxygen using ambient hydrogen as the only ionizing medium at or near the termination shock are able to successfully model this transition. The simulated oxygen intensity, however, appears deficient at high energies, where multiply-charged ACRs dominate. This paper presents further simulations that now include neutral helium as part of the ionizing medium in addition to the ambient, neutral hydrogen. The inclusion of helium helps reduce the deficiency, but appears to fall short of accounting fully for the observed spectrum. To that end, including heavier neutrals, e.g., oxygen, as well as taking multi-electron stripping into account, are suggested for more realistic modeling of the observed charge states of ACRs.