We present a suite of 15 cosmological zoom-in simulations of isolated dark matter halos, all with masses of $M_{\rm halo} ≈ 10^{10}\,{\rm M}_⊙$ at $z=0$, in order to understand the relationship between halo assembly, galaxy formation, and feedback's effects on the central density structure in dwarf galaxies. These simulations are part of the Feedback in Realistic Environments (FIRE) project and are performed at extremely high resolution. The resultant galaxies have stellar masses that are consistent with rough abundance matching estimates, coinciding with the faintest galaxies that can be seen beyond the virial radius of the Milky Way ($M_⋆/{\rm M}_⊙≈ 10^5-10^7$). This non-negligible spread in stellar mass at $z=0$ in halos within a narrow range of virial masses is strongly correlated with central halo density or maximum circular velocity $V_{\rm max}$. Much of this dependence of $M_⋆$ on a second parameter (beyond $M_{\rm halo}$) is a direct consequence of the $M_{\rm halo}∼10^{10}\,{\rm M}_⊙$ mass scale coinciding with the threshold for strong reionization suppression: the densest, earliest-forming halos remain above the UV-suppression scale throughout their histories while late-forming systems fall below the UV-suppression scale over longer periods and form fewer stars as a result. In fact, the latest-forming, lowest-concentration halo in our suite fails to form any stars. Halos that form galaxies with $M_⋆\gtrsim2\times10^{6}\,{\rm M}_⊙$ have reduced central densities relative to dark-matter-only simulations, and the radial extent of the density modifications is well-approximated by the galaxy half-mass radius $r_{1/2}$. This apparent stellar mass threshold of $M_⋆ ≈ 2\times 10^{6} ≈ 2\times 10^{-4} \,M_{\rm halo}$ is broadly consistent with previous work and provides a testable prediction of FIRE feedback models in LCDM. ; Comment: 16 pages, 16 figures; submitted to MNRAS