Among the different types of bilayer tunneling field-effect transistors exploiting interband tunneling phenomena with tunneling directions aligned with gate-induced electric fields, the utilization of InAs/GaSb channels proves to be an appealing means to enhance ON-current levels. Ultrathin channel thicknesses make quantum confinement be the agent that closes the broken gap of the InAs/GaSb heterojunction leading to a staggered gap which blocks the tunneling current in the OFF state. In this paper, the gate leakage tunneling current is analyzed as one of the main critical processes degrading the performance of the proposed structure. Appropriate gate stacks of HfO2/Al2O3 combined with gate-to-drain underlaps are shown to effectively suppress this leakage tunneling, while at the same time, preserve an adequate electrostatic control over the channel. Simulation results for the most optimized configurations feature ON-state levels of up to 400 mu A/mu m and subthreshold swings of approximate to 3 mV/dec over more than 7 decades of current.