We present a suite of FIRE-2 cosmological zoom-in simulations of isolated field dwarf galaxies, all with masses of $M_mathrm{halo} approx 10^{10},$M$_odot$ at $z=0$, across a range of dark matter models. For the first time, we compare how both self-interacting dark matter (SIDM) and/or warm dark matter (WDM) models affect the assembly histories as well as the central density structure in fully hydrodynamical simulations of dwarfs. Dwarfs with smaller stellar half-mass radii (r$_{1/2}<500$ pc) have lower $sigma_star/V_mathrm{max}$ ratios, reinforcing the idea that smaller dwarfs may reside in halos that are more massive than is naively expected. The majority of dwarfs simulated with self-interactions actually experience contraction of their inner density profiles with the addition of baryons relative to the cores produced in dark-matter-only runs, though the simulated dwarfs are always less centrally dense than in $Lambda$CDM. The V$_{1/2}-$r$_{1/2}$ relation across all simulations is generally consistent with observations of Local Field dwarfs, though compact objects such as Tucana provide a unique challenge. Spatially-resolved rotation curves in the central regions ($<400$ pc) of small dwarfs could provide a way to distinguish between CDM, WDM, and SIDM, however: at the masses probed in this simulation suite, cored density profiles in dwarfs with small r$_{1/2}$ values can only originate from dark matter self-interactions.
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