A nuclear mass survey of rare-earth isotopes has been conducted with the Canadian Penning Trap mass spectrometer using the most neutron-rich nuclei thus far extracted from the CARIBU facility. We present a collection of 12 nuclear masses determined with a precision of $\ensuremath{\le}10$ keV/$c{}^{2}$ for $Z=58--63$ nuclei near $N=100$. Independently, a detailed study exploring the role of nuclear masses in the formation of the $r$-process rare-earth abundance peak has been performed. Employing a Markov chain Monte Carlo (MCMC) technique, mass predictions of lanthanide isotopes have been made which uniquely reproduce the observed solar abundances near $A=164$ under three distinct astrophysical outflow conditions. We demonstrate that the mass surface trends thus far mapped out by our measurements are most consistent with MCMC mass predictions given an $r$ process that forms the rare-earth peak during an extended $(n,\ensuremath{\gamma})\ensuremath{\rightleftarrows}(\ensuremath{\gamma},n)$ equilibrium.