Wheeled bipedal robots have the capability to execute agile and versatile locomotion tasks in unknown terrains, with balancing being a key criteria in evaluating their dynamic performance. This paper focuses on enhancing the balancing performance of wheeled bipedal robots through innovations in both hardware and software aspects. A bio-inspired mechanical design, inspired by the human barbell squat, is proposed and implemented to achieve an efficient distribution of load onto the limb joints. This design improves knee torque joint efficiency and facilitates control over the distribution of the center of mass (CoM). Meanwhile, a customized balance model, namely the wheeled linear inverted pendulum (wLIP), is developed. The wLIP surpasses other alternatives by providing a more accurate estimation of wheeled robot dynamics while ensuring balancing stability. Experimental results demonstrate that the robot is capable of maintaining balance while manipulating pelvis states and CoM velocity; furthermore, it exhibits robustness against external disturbances and unknown terrains.