Abstract:We investigate the spatially-resolved ISM properties of the local ($z = 0.048$), IR-luminous ($L_{\rm IR} = 10^{11.87}$\,L$_\odot$), late-stage galaxy merger IRAS F01364-1042, combining multi-wavelength IFU observations from \textit{JWST/MIRI-MRS}, ALMA and Keck/KCWI. Using these datasets, we construct emission line maps of several key tracers of the ionized (e.g., [Ne\,II]\,12.8$\mu$m, [O\,III]$\lambda5007$), warm molecular (e.g., \ce{H2}\,0-0\,S(3)), and cold molecular gas (e.g., CO (J$=2-1$)), and perform detailed decomposition of spectra extracted in resolved regions across the areas of emission. We confirm the presence of a multi-phase galactic biconical outflow along the minor axis of a highly inclined rotating disk. The inferred outflow velocities are $\sim\,$500 - 600\,km\,s$^{-1}$ $\sim\,$350\,km\,s$^{-1}$, and $\sim\,$200 - 300\,km\,s$^{-1}$, in the ionized, warm and cold molecular phase, respectively, with corresponding mass outflow rates of $\sim 0.3 - 2.3$, $\sim 31$, and $\sim 38 - 240$\,M$_\odot\,$yr$^{-1}$. The cold molecular phase dominates both the total mass outflow rate and the associated kinetic energy ($\sim\,2 - 8 \times 10^{42}$\,erg\,s$^{-1}$). We confirm, for the first time, a dust-obscured AGN in IRAS\,F01364-1042, via detection of the [Ne\,V]\,14.3$\mu$m line. The low inferred AGN bolometric luminosity ($1.2 - 1.8 \times 10^{43}$\,erg\,s$^{-1}$) suggests that the nuclear starburst alone, with a star formation rate of $\sim 40 - 60$\,M$_\odot$\,yr$^{-1}$, can account for the energy required to drive the outflow, though a more active AGN phase in the recent past may have also played a role. Our work showcases the necessity of multi-wavelength observations for interpreting the gas dynamics in merger-driven dusty starbursts, and the capability of \textit{JWST/MIRI-MRS} to uncover obscured, low-luminosity AGN that may be common in these systems.