Motion planning problems for physically-coupled multi-robot systems in cluttered environments are challenging due to their high dimensionality. Existing methods combining sampling-based planners with trajectory optimization produce suboptimal results and lack theoretical guarantees. We propose Physically-coupled discontinuity-bounded Conflict-Based Search (pc-dbCBS), an anytime kinodynamic motion planner, that extends discontinuity-bounded CBS to rigidly-coupled systems. Our approach proposes a tri-level conflict detection and resolution framework that includes the physical coupling between the robots. Moreover, pc-dbCBS alternates iteratively between state space representations, thereby preserving probabilistic completeness and asymptotic optimality while relying only on single-robot motion primitives. Across 25 simulated and six real-world problems involving multirotors carrying a cable-suspended payload and differential-drive robots linked by rigid rods, pc-dbCBS solves up to 92% more instances than a state-of-the-art baseline and plans trajectories that are 50-60% faster while reducing planning time by an order of magnitude.