We theoretically investigate circular dichroism (CD) in second- and third-harmonic generation (SHG and THG) in the chiral topological semimetal CoSi. We demonstrate that both SHG and THG exhibit dichroic responses of order unity, while their robustness against spectral broadening is strikingly different. Specifically, while SHG-CD is strongly suppressed by dissipation, THG-CD remains robust over a wide frequency range. We show that this qualitative difference originates from the phase structure of the nonlinear current, where SHG-CD arises from subleading interference processes that are sensitive to dephasing, whereas THG-CD emerges already at the leading nonlinear order and is therefore protected against spectral broadening. As a result, THG-CD provides a robust probe of chirality encoded in nonequilibrium electronic dynamics. We further reveal non-monotonic frequency dependences and pronounced sensitivity of harmonic emission to the polarization state and crystallographic orientation of the driving field. Our results provide a representative example of robust nonlinear chiroptical responses in chiral topological semimetals and highlight the potential of high-harmonic spectroscopy as a probe of phase-resolved electronic dynamics.