Abstract We present multiwavelength time-series spectroscopy of SN 2013aa and SN 2017cbv, two Type Ia supernovae (SNe Ia) on the outskirts of the same host galaxy, NGC 5643. This work utilizes new nebular-phase near-infrared (NIR) spectra obtained by the Carnegie Supernova Project-II, in addition to previously published optical and NIR spectra. Using nebular-phase [Fe ii] lines in the optical and NIR, we examine the explosion kinematics and test the efficacy of several common emission-line-fitting techniques. The NIR [Fe ii] 1.644 μm line provides the most robust velocity measurements against variations due to the choice of the fit method and line blending. The resulting effects on velocity measurements due to choosing different fit methods, initial fit parameters, continuum and line profile functions, and fit region boundaries were also investigated. The NIR [Fe ii] velocities yield the same radial shift direction as velocities measured using the optical [Fe ii] λ7155 line, but the sizes of the shifts are consistently and substantially lower, pointing to a potential issue in optical studies. The NIR [Fe ii] 1.644 μm emission profile shows a lack of significant asymmetry in both SNe, and the observed low velocities elevate the importance for correcting for any velocity contribution from the host galaxy’s rotation. The low [Fe ii] velocities measured in the NIR at nebular phases disfavor progenitor scenarios in close double-degenerate systems for both SN 2013aa and SN 2017cbv. The time evolution of the NIR [Fe ii] 1.644 μm line also indicates moderately high progenitor white dwarf central density and potentially high magnetic fields.