A primary goal of the upcoming Deep Underground Neutrino Experiment (DUNE) is to measure the neutrinos produced by a Galactic core-collapse supernova if one should occur during the lifetime of the experiment. The liquid-argon-based detectors planned for DUNE are expected to be uniquely sensitive to the component of the supernova flux, enabling a wide variety of physics and astrophysics measurements. A key requirement for a correct interpretation of these measurements is a good understanding of the energy-dependent total cross section σ(E) for charged-current absorption on argon. In the context of a simulated extraction of supernova spectral parameters from a toy analysis, we investigate the impact of σ(E) modeling uncertainties on DUNE's supernova neutrino physics sensitivity for the first time. We find that the currently large theoretical uncertainties on σ(E) must be substantially reduced before the flux parameters can be extracted reliably; in the absence of external constraints, a measurement of the integrated neutrino luminosity with less than 10% bias with DUNE requires σ(E) to be known to about 5%. The neutrino spectral shape parameters can be known to better than 10% for a 20% uncertainty on the cross-section scale, although they will be sensitive to uncertainties on the shape of σ(E). A direct measurement of low-energy -argon scattering would be invaluable for improving the theoretical precision to the needed level.