I recently reported [Phys. Rev. A 51, 2222 (1995)] on simulations of charge-exchange and excitation processes in collisions between a helium nucleus and a ground-state hydrogen atom that were performed in the multichannel perturbed-stationary-state framework without employing electron translation factors [Phys. Rev. A 51, 2199 (1995)]. A simulation with 45 adiabatic molecular orbitals reproduced the measured cross sections for electron capture and for the ensuing Lyman fluorescence lines of ${\mathrm{He}}^{+},$ up to the collisional ionization threshold $(\ensuremath{\sim}9\mathrm{k}\mathrm{e}\mathrm{V}/\mathrm{a}\mathrm{m}\mathrm{u}).$ In this study, the collision of ${\mathrm{He}}^{2+}$ on $\mathrm{H}(1s)$ is investigated further using a multichannel propagator defined over the former perturbed-stationary-state basis extended to include 35 ${L}^{2}$ ionization pseudostates of the linear combination of atomic orbitals (LCAO) type. The simulated total charge-transfer cross section is now in good agreement with experiment across the peak plateau and well into the falloff wing, where it begins to tail away at energies above 30 keV/amu because the LCAO ionization set is deficient. Concurrently, throughout the range where collisional ionization is important, state-selective cross sections are improved compared to our previous study. Approximately 25% of the calculated peak transfer cross section is associated with $n>~3{\mathrm{He}}^{+}$ levels. The multichannel propagator simulations imply that (i) decay cascades contribute almost one-third of the ensuing spontaneous He II Lyman-\ensuremath{\alpha} fluorescence cross section [J. Phys. B 24, 4025 (1991)] that is accurately reproduced and (ii) capture-induced Balmer-\ensuremath{\alpha} and Paschen-\ensuremath{\alpha} emissions are considerably stronger than the values either measured in the aforementioned experiment or computed by translation-factor models.