Doppler-limited, high-precision, molecular spectroscopy in the linear regime of interaction may refine our knowledge of the Boltzmann constant. To this end, the global uncertainty in the retrieval of the Doppler width should be reduced down to 1 part over 106, which is a rather challenging target. So far, Doppler-broadening thermometry has been mostly limited by the uncertainty associated to the line shape model that is adopted for the nonlinear least-squares fits of experimental spectra. In this paper, we deeply investigate this issue by using a very realistic and sophisticated model, known as partially correlated speed-dependent Keilson-Storer profile, to reproduce near-infrared water spectra. A global approach has been developed to fit a large number of numerically simulated spectra, testing a variety of simplified line-shape models. It turns out that the most appropriate model is the speed-dependent hard-collision profile. We demonstrate that the Doppler width can be determined with relative precision and accuracy, respectively, of 0.42 and 0.75 part per million.