Mobile manipulators typically encounter significant challenges in navigating narrow, cluttered environments due to their high-dimensional state spaces and complex kinematics. While reactive methods excel in dynamic settings, they struggle to efficiently incorporate complex, coupled constraints across the entire state space. In this work, we present a novel local reactive controller that reformulates the time-domain single-step problem into a multi-step optimization problem in the spatial domain, leveraging the propagation of a serial kinematic chain. This transformation facilitates the formulation of customized, decoupled link-specific constraints, which is further solved efficiently with augmented Lagrangian differential dynamic programming (AL-DDP). Our approach naturally absorbs spatial kinematic propagation in the forward pass and processes all link-specific constraints simultaneously during the backward pass, enhancing both constraint management and computational efficiency. Notably, in this framework, we formulate collision avoidance constraints for each link using accurate geometric models with extracted free regions, and this improves the maneuverability of the mobile manipulator in narrow, cluttered spaces. Experimental results showcase significant improvements in safety, efficiency, and task completion rates. These findings underscore the robustness of the proposed method, particularly in narrow, cluttered environments where conventional approaches could falter. The open-source project can be found at https://github.com/Chunx1nZHENG/MM-with-Whole-Body-Safety-Release.git.