Neutron star mergers (NSMs) are promising astrophysical sites for the rapid neutron-capture ("r") process, but can their integrated yields explain the majority of heavy-element material in the Galaxy? One method to address this question implements a forward approach that propagates NSM rates and yields along with stellar formation rates and compares those results with observed chemical abundances of r-process-rich, metal-poor stars. In this work, we take the inverse approach by utilizing r-process-element abundance ratios of metal-poor stars as input to reconstruct the properties—especially the masses—of their neutron star (NS) binary progenitors. This novel analysis provides an independent avenue for studying the population of the original NS binary systems that merged and produced the r-process material now incorporated in Galactic metal-poor halo stars. We use ratios of elements typically associated with the limited-r-process and the actinide region to those in the lanthanide region (i.e., Zr/Dy and Th/Dy) to probe the NS masses of the progenitor merger. We find that NSMs can account for all r-process material in metal-poor stars that display r-process signatures, while simultaneously reproducing the present-day distribution of double-NS systems. Notably, with our model assumptions and the studied stellar sample, we postulate that the most r-process enhanced stars (the r–II stars) on their own would require progenitor NSMs of asymmetric systems that are distinctly different from present ones in the Galaxy. We also explore variations to the model and find that the predicted degree of asymmetry is most sensitive to the electron fraction of the remnant disk wind.