The flux of cosmic rays to the atmosphere has been reported to correlate with cloud and aerosol properties. One proposed mechanism for these correlations is the "ion-aerosol clear-air" mechanism where the cosmic rays modulate atmospheric ion concentrations, ion-induced nucleation of aerosols and cloud condensation nuclei (CCN) concentrations. We use a global chemical transport model with online aerosol microphysics to explore the dependence of CCN concentrations on the cosmic-ray flux. Expanding upon previous work, we test the sensitivity of the cosmic-ray/CCN connection to several uncertain parameters in the model including primary emissions, Secondary Organic Aerosol (SOA) condensation and charge-enhanced condensational growth. The sensitivity of CCN to cosmic rays increases when simulations are run with decreased primary emissions, but show location-dependent behavior from increased amounts of secondary organic aerosol and charge-enhanced growth. For all test cases, the change in the concentration of particles larger than 80 nm between solar minimum (high cosmic ray flux) and solar maximum (low cosmic ray flux) simulations is less than 0.2 %. The change in the total number of particles larger than 10 nm was larger, but always less than 1 %. The simulated change in the column-integrated Ångström exponent was negligible for all test cases. Additionally, we test the predicted aerosol sensitivity to week-long Forbush decreases of cosmic rays and find that the maximum change in aerosol properties for these cases is similar to steady-state aerosol differences between the solar maximum and solar minimum. These results provide evidence that the effect of cosmic rays on CCN and clouds through the ion-aerosol clear-sky mechanism is limited by dampening from aerosol processes.