Planning the motion path for a tightly coupled dual-arm space manipulator under closed-chain constraints is a fundamental yet challenging problem in on-orbit assembly of large-scale space structures. The closed-chain constraints significantly reduce the feasible configuration space, making it difficult for existing planners to efficiently generate collision-free motions, especially in cluttered environments. To address this issue, this paper proposes a task-space constrained bidirectional rapidly-exploring random tree algorithm, termed TCBiRRT. Unlike conventional methods that operate in the high-dimensional configuration space, the proposed approach performs random sampling and node expansion directly in the task space defined by the manipulated object pose. A task-space node expansion strategy is developed to generate candidate object motions, which are then mapped to continuous joint paths using a path inverse kinematics algorithm. The method is further integrated with a bidirectional RRT framework and a regrasp mechanism to efficiently connect two random trees. Extensive simulations are conducted in representative on-orbit assembly scenarios with varying levels of environmental complexity. The results demonstrate that TCBiRRT achieves significantly higher success rates and orders-of-magnitude improvements in planning time compared to state-of-the-art planners. The proposed method provides an efficient and robust solution for motion planning of tightly coupled dual-arm space manipulators.