We present new ALMA Band 7 ($sim340$ GHz) observations of the dense gas tracers HCN, HCO$^+$, and CS in the local, single-nucleus, ultraluminous infrared galaxy IRAS 13120-5453. We find centrally enhanced HCN (4-3) emission, relative to HCO$^+$ (4-3), but do not find evidence for radiative pumping of HCN. Considering the size of the starburst (0.5 kpc) and the estimated supernovae rate of $sim1.2$ yr$^{-1}$, the high HCN/HCO$^+$ ratio can be explained by an enhanced HCN abundance as a result of mechanical heating by the supernovae, though the active galactic nucleus and winds may also contribute additional mechanical heating. The starburst size implies a high $Sigma_{IR}$ of $4.7times10^{12}$ $L_{odot}$ kpc$^{-2}$, slightly below predictions of radiation-pressure limited starbursts. The HCN line profile has low-level wings, which we tentatively interpret as evidence for outflowing dense molecular gas. However, the dense molecular outflow seen in the HCN line wings is unlikely to escape the galaxy and is destined to return to the nucleus and fuel future star formation. We also present modeling of Herschel observations of the H$_2$O lines and find a nuclear dust temperature of $sim40$ K. IRAS 13120-5453 has a lower dust temperature and $Sigma_{IR}$ than is inferred for the systems termed compact obscured nuclei (such as Arp 220 and Mrk 231). If IRAS 13120-5453 has undergone a compact obscured nucleus phase, we are likely witnessing it at a time when the feedback has already inflated the nuclear ISM and diluted star formation in the starburst/AGN core.
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