Dual quaternions offer an elegant, singularity-free mechanism to describe the coupled 6-DOF dynamics essential for close-proximity operations. By effectively capturing complex attitude-orbit interactions, this framework provides a robust mathematical foundation superior to conventional separate modeling techniques. However, existing dual quaternion-based control strategies typically rely on fixed feedback parameters. Such static configurations inherently lack the flexibility to handle the dynamic time-varying nature of space missions. When facing large initial tracking errors, parameter uncertainties, or external disturbances, fixed-gain controllers often struggle to balance conflicting performance requirements, such as rapid convergence versus overshoot suppression. This limitation frequently results in performance degradation, including reduced control accuracy and decreased robustness, thereby constraining their adaptability in complex autonomous rendezvous scenarios. To address these limitations, this paper proposes a novel Predictive Spider Wasp Optimization-based PD-like controller(PSWO-PD). By integrating dual quaternion control with a bio-inspired Predictive Spider Wasp Optimization(PSWO) algorithm, the system moves beyond traditional fixed-gain structures and adopts a receding horizon approach for real-time adaptation. First, the nonlinear coupled dynamic model of the spacecraft is established using dual quaternions, and the relative motion dynamics are derived to serve as the prediction model. Subsequently, a PD-like controller is developed within the PSWO framework. Unlike standard heuristic tuning methods, the PSWO algorithm mimics the unique hunting and nesting behaviors of spider wasps. This mechanism provides an efficient balance between exploration and exploitation, enabling fast convergence to optimal solutions. In each control period, the algorithm utilizes the dynamic model to predict state evolution over a finite future horizon. It searches for optimal gain matrices to minimize a composite cost function. This function explicitly penalizes predicted translational and rotational tracking errors as well as control effort. This method effectively transforms the traditional PD controller into a predictive optimizer, capable of dynamically adjusting response characteristics based on the instantaneous spacecraft state. For instance, gains are increased to ensure fast tracking when errors are large, and decreased to avoid overshoot as the spacecraft approaches the target. Furthermore, the stability of the time-varying control law is rigorously proven based on passivity theory. Although feedback parameters are modulated online, the optimization search space is strictly constrained to ensure parameter strictly positive definiteness. This constraint guarantees that the controller behaves physically as a dissipative system with variable stiffness and damping, which further guarantees global asymptotic stability. Finally, numerical simulations are performed for a close-proximity mission under both nominal and disturbance conditions. Comparative results against a traditional fixed-gain PD-like controller validate the effectiveness, superior control accuracy, and robustness of the proposed PSWO-PD algorithm.