We utilize James Webb Space Telescope/Mid Infrared Instrument (JWST/MIRI) Integral Field Unit observations from the Galaxy Activity, Torus and Outflow Survey to investigate the diverse range of ionized outflow rates of obscured active galactic nuclei (AGN) with similar bolometric luminosity and explore potential associations with AGN feedback. We explore spatial correlations between ionized emission potentially associated with fast shocks ([Fe II]5.34μm) and the excitation of H2. We further constrain our investigation to the inner 400 pc (the nuclear and circumnuclear regions r < 200 pc), and estimate the excitation temperature and column density of H2 assuming local thermodynamic equilibrium and using the S(1)─S(8) rotational H2 emission lines visible to JWST/MIRI spectroscopy. We report the molecular gas temperature of the deprojected 400 pc nuclear region to correlate with the ionized mass outflow rate. We also observe a stronger degree of spatial correlation between [Fe II]5.34μm emission and H2 gas temperature. We observe regions of enhanced [Fe II]5.34μm/[Ar II]6.99μm spatially coincident with the ionization cones of objects with higher ionized outflow rates and [Fe II]5.34μm/[Ar II]6.99μm in the deprojected 400 pc nuclear region to scale positively with both the ionized outflow rate and the estimated molecular gas temperature. We do not observe the estimated jet cavity power within the central 400 pc as strongly correlated with the ionized mass outflow rate or molecular gas temperature of the nuclear region. We take the preceding observations to suggest a higher degree of interaction between AGN outflows and the circumnuclear disk.