We present a detailed study of the inner regions of NGC 7582, a nearby Seyfert 2 galaxy, from the Galaxy Activity, Torus, and Outflow Survey (GATOS). The galaxy hosts a circumnuclear star-forming disc and an active galactic nucleus (AGN)-driven biconical ionized outflow. Using James Webb Space Telescope Near-Infrared Spectrograph (NIRSpec) and Mid-Infrared Instrument/Medium-Resolution Spectrometer (MIRI/MRS) integral-field spectroscopy, we analyse ionic emission lines spanning a wide range of ionization potentials (IPs, –126 eV). Gaussian line-profile fitting reveals kinematic stratification: low-IP species ( eV; e.g. [Fe ii], [Ar ii], and [Ne ii]) trace ordered disc rotation with PA , while high-IP species ( eV; e.g. [O iv], [Mg iv], and [Ne v]) follow the outflow with PA . Outflowing gas exhibits systematically higher velocity dispersions ( km s−1) than the disc ( km s−1), consistent with turbulent or bulk motions. Intermediate-IP lines, [S iii], [Ar iii], and [Ne iii], show contributions from both components, with the outflow characterized by higher dispersion, lower amplitude, and higher velocities in double-Gaussian fits. For these lines, a thin inclined disc plus 1D outflow model enables robust separation and quantification of the disc and outflow velocity fields. The outflow is consistent with a hollow bicone capable of accelerating gas beyond the local escape velocity, implying most material is unlikely to be re-accreted. The ionization cone opening angle shows no dependence on IP, indicating the AGN torus polar regions are largely unobscured. Our study provides new insights into AGN-driven outflows and circumnuclear disc dynamics, offering a framework to disentangle overlapping interstellar medium kinematics in nearby active galaxies.