Accretion disks in active galactic nuclei (AGN) are expected to host embedded stellar-mass black holes (BHs) whose gas capture rates can be highly super-Eddington. Without feedback, they would rapidly grow and deplete the disk, but at such high feeding rates, radiation is advected inward with the flow. We test whether the resulting accretion-powered jet and its shocked cocoon can (i) self-regulate the BH's growth rate and (ii) break out of the AGN disk. Using GIZMO, we perform three-dimensional simulations of a 10 rm M_⊙ BH, embedded in dense AGN-disk gas, launching a particle-spawned jet whose mass flux scales with the accretion rate. We explore a range of jet velocities, ambient temperatures and densities, and three cooling regimes. We find that the jet-inflated cocoon stalls near the Bondi radius and maintains a modest aspect ratio and the self-regulated mass flux is determined by momentum balance at the Bondi radius. Cooling controls the mode of self-regulated accretion, from large-amplitude oscillation cycles at low metallicity to weak, quasi-steady modulation at solar metallicity. Scaling our results to outer disk conditions implies that mechanical feedback limits BH growth to within an order of magnitude of the Eddington rate – largely preventing the overgrowth problem. Over the same disk region, the cocoon remains confined within the disk. In the inner disk, accretion onto the embedded BHs is limited by tidal shear rather than the Bondi radius, and our simulations no longer apply. Whether runaway growth and cocoon breakout remains possible in these inner regions needs further study.