Human-AI cooperative classification (HAI-CC) aims to develop hybrid intelligent systems that enhance decision-making in various high-stakes real-world scenarios by leveraging both human expertise and AI capabilities. Current HAI-CC methods primarily focus on learning-to-defer (L2D), where decisions are deferred to human experts when AI is not confident, and learning-to-complement (L2C), where AI and human experts make predictions cooperatively. However, existing research in both L2D and L2C has not effectively been explored under diverse expert knowledge to improve decision-making, particularly when constrained by the operation cost of human involvement. In this paper, we address this research gap by proposing the Coverage-constrained Learning to Defer and Complement with Specific Experts (CL2DC) method. In particular, CL2DC assesses input data before making final decisions through either AI prediction alone or by deferring to or complementing a specific human expert. Furthermore, we propose a coverage-constrained optimisation to control the cooperation cost, ensuring it approximates a target probability for AI-only selection. This approach enables an effective assessment of system performance within a specified budget. Comprehensive evaluations on both synthetic and real-world datasets demonstrate that CL2DC achieves superior performance compared to state-of-the-art HAI-CC methods.