Inspired by sled-pulling dogs in transportation, we present a cable-trailer integrated with a quadruped robot system. The motion planning of this system faces challenges due to the interactions between the cable's state transitions, the trailer's nonholonomic constraints, and the system's underactuation. To address these challenges, we first develop a hybrid dynamics model that captures the cable's taut and slack states. A search algorithm is then introduced to compute a suboptimal trajectory while incorporating mode transitions. Additionally, we propose a novel collision avoidance constraint based on geometric polygons to formulate the trajectory optimization problem for the hybrid system. The proposed method is implemented on a Unitree A1 quadruped robot with a customized cable-trailer and validated through experiments. The real system demonstrates both agile and safe motion with cable mode transitions.