Operating at high temperatures in the range of thermoelectric coolers is essential for terahertz quantum cascade lasers to real applications. The use of scattering-assisted injection scheme enables an increase in operating temperature. This concept, however, has not been implemented in a short-period structure consisting of two quantum wells. In this work, based on non-equilibrium Green's function calculations, it emphasizes on the current leakage and parasitic absorption via high-energy states as fundamental limitations in this scheme with short-period. A new design concept employing asymmetric wells composition is proposed to suppress these limitations. A peak gain of 40 cm-1 at 230 K is predicted in the GaAs/AlGaAs semiconductor material system with an emission frequency of 3.5 THz.