||The first generation of galaxies is expected to form in minihaloes, accreting gas through H-2 cooling, and possessing unique properties. Although unlikely to be directly detected in UV/infrared surveys, the radiation from these molecular-cooling galaxies (MCGs) could leave an imprint in the 21-cm signal from the Cosmic Dawn. Here, we quantify their detectability with upcoming radio interferometers. We generate mock 21-cm power spectra using a model for both MCGs as well as more massive, atomic-cooling galaxies, allowing both populations to have different properties and scaling relations. The galaxy parameters are chosen so as to be consistent with: (i) high-redshift UV luminosity functions; (ii) the upper limit on the neutral fraction from QSO spectra; (iii) the Thomson scattering optical depth to the CMB; and (iv) the timing of the recent putative EDGES detection. The latter implies a significant contribution of MCGs to the Cosmic Dawn, if confirmed to be cosmological. We then perform Bayesian inference on two models including and ignoring MCG contributions. Comparing their Bayesian evidences, we find a strong preference for the model including MCGs, despite the fact that it has more free parameters. This suggests that if MCGs indeed play a significant role in the Cosmic Dawn, it should be possible to infer their properties from upcoming 21-cm power spectra. Our study illustrates how these observations can discriminate among uncertain galaxy formation models with varying complexities, by maximizing the Bayesian evidence.