The temporal-domain topology optimization of transient dynamic problems is of paramount importance in engineering, as it plays a crucial role in preventing potential structural failures in the presence of dynamic loading conditions. To improve the quality of optimization outcomes, integrating both stiffness and strength as concurrent performance metrics within the optimization framework has to be considered. However, optimization of these problems poses core challenges, primarily due to their intricate nature and the risk of convergence instability. On the one hand, the coupling between dynamic structural response and stress limitations introduces significant nonlinearity. On the other hand, penalty-based approaches suffer from ill-conditioning when confronted with simultaneous volume and stress restrictions, rendering parameter selection dependent on trial-and-error calibration to circumvent numerical instabilities, high frequency spurious oscillations, artificial damping artifacts, and phase errors impacting sensitivity calculations. In response to these challenges, this work proposes an Enhanced Extended Augmented Lagrangian Method (ALM) that effectively converts intricate multi-constrained problems into unconstrained ones aiming to minimize the total transient strain energy while ensuring compliance with stress and volume constraints. To enable the efficient application of the Moving Asymptote Method for solving a series of subproblems, a Ψ-step iterative strategy is introduced to ascertain the optimal initial values of the ALM parameters and a piecewise function for the gradual tuning of penalty factors. This approach improves the stability and precision of the proposed algorithm. The performence of this method is demonstrated through comparative experiments conducted on established benchmark problems.