This article investigates the cooperative rendezvous control problem for perturbed heterogeneous marine systems composed of an autonomous underwater vehicle (AUV) and an autonomous surface vehicle (ASV). A novel Lyapunov-based model predictive control (LMPC) framework is presented to accomplish safe and precise rendezvous under input limitations and external disturbances. First, by incorporating the prescribed performance control (PPC) technique into the LMPC framework, we transform the original ascending state of the AUV into a self-constrained state, which serves as the decision variable of the model predictive control (MPC) optimization problem. Then, PPC-aided auxiliary control laws based on disturbance observers (DOBs) are designed to establish a robust contractive constraint to provide stability margins. Combining the LMPC with the PPC technique makes the original state-constrained problem an equivalent state-constraint-free problem. By addressing the MPC problem for the equivalent unconstrained system, the proposed method preserves the rendezvous safety. With the robust contractive constraint, the proposed safety-preserving LMPC (SP-LMPC) controller can inherit robustness and stability from the robust auxiliary control laws. Furthermore, theoretical analyses are conducted to assess recursive feasibility and closed-loop stability. With comprehensive theoretical support, the proposed method provides a new framework to simultaneously address state constraints and disturbances for highly nonlinear marine systems. Finally, simulations and comparisons are conducted to demonstrate the effectiveness and advantages of the proposed algorithm.