Studies aiming to determine the astrophysical origins of nuclei produced by the rapid neutron capture process ($r$ process) rely on nuclear properties as inputs for simulations. The solar abundances can be used as a benchmark for such calculations, with the $r$-process rare-earth peak (REP) around mass number ($A$) 164 being of special interest due to its presently unknown origin. With the advancement of rare isotope beam production over the last decade and improvement in experimental sensitivities, many of these REP nuclides have become accessible for measurement. Masses are one of the most critical inputs as they impact multiple nuclear properties, namely the neutron-separation energies, neutron capture rates, $\beta$-decay rates, and $\beta$-delayed neutron emission probabilities. In this work, we report masses of 20 neutron-rich nuclides (along the Ba, La, Ce, Pr, Nd, Pm, Gd, Dy and Ho isotopic chains) produced from ${}^{252}$Cf fission at the CAlifornium Rare Isotope Breeder Upgrade (CARIBU) facility at Argonne National Laboratory. The masses were measured with the Canadian Penning trap (CPT) mass spectrometer using the Phase-Imaging Ion-Cyclotron-Resonance (PI-ICR) technique. These include the first reported mass of ${}^{168}$Gd, first directly measured mass of an isomer of ${}^{168}$Ho, eighteen improvements on current literature values, and masses of some of the most weakly produced isotopes from ${}^{252}$Cf fission, belonging to 10${}^{-5}$% and 10${}^{-4}$% fission branches. To explore the impact of these measurements in astrophysics, we first directly apply the masses newly reported in this work in an $r$-process calculation and show that these mildly affect rare-earth abundances on the left side of the peak. We then use these new masses along with previously published CPT masses to inform predictions for a Markov Chain Monte Carlo (MCMC) procedure aiming to identify the astrophysical conditions consistent with both Solar data and mass measurements. As this work presents the final measurements of rare-earth nuclei that will be performed by the CPT at CARIBU, we culminate the partnering of CPT masses and our MCMC method by exploring how our predictions respond when the full set of CPT measurements is incorporated into our procedure. We show that the MCMC responds to this updated mass information, producing refined results for both mass predictions and REP abundances.