We introduce a gray-box enhanced local search for k-bounded multi-objective optimization that combines decomposition and cooperation through a flexible, plug-and-play design. We study the management of the gray-box score vector and investigate strategies for designing cooperation, assessing their impact on both approximation quality and computational efficiency. Our experimental evaluation considers large-scale random and adjacent NK landscapes with varying ruggedness, providing a comprehensive comparison of the cooperative modes in terms of solution quality and CPU time. Beyond substantially speeding up the search compared to a black-box approach, our empirical analysis demonstrates the critical role of effective score vector management and provides insights on which cooperative designs achieve the best trade-off between efficiency and quality in gray-box multi-objective optimization.