Abstract 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 disk and an AGN-driven biconical ionised outflow. Using JWST NIRSpec and MIRI/MRS integral-field spectroscopy, we analyse ionic emission lines spanning a wide range of ionisation potentials (IPs, ∼8–126 eV). Gaussian line-profile fitting reveals kinematic stratification: low-IP species (≲ 20 eV; e.g., [Fe II], [Ar II], [Ne II]) trace ordered disk rotation with PA ∼−12 ± 3○, while high-IP species (≳ 35 eV; e.g., [O IV], [Mg IV], [Ne V]) follow the outflow with PA ∼54 ± 10○. Outflowing gas exhibits systematically higher velocity dispersions (119 ± 13 km/s) than the disk (78 ± 11 km/s), consistent with turbulent or bulk motions. Intermediate-IP lines, [S III], [Ar III], and [Ne III], show contributions from both components, with the outflow characterised by higher dispersion, lower amplitude, and higher velocities in double-Gaussian fits. For these lines, a thin inclined disk plus one-dimensional outflow model enables robust separation and quantification of the disk 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 ionisation 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 disk dynamics, offering a framework to disentangle overlapping ISM kinematics in nearby active galaxies.