Construction machinery operations are intrinsically linked to critical societal challenges, including safety risks and carbon emissions. In response to the high incidence of fatal accidents during installation and dismantling phases, the Chinese government has officially promoted integrated installation–dismantling services to enhance construction safety since 2023. However, the economic viability of this policy for leasing companies remains largely underexplored. To address this gap, this paper develops a leasing-oriented closed-loop construction machinery supply chain model that incorporates integrated installation–dismantling services under an industrial internet platform. The study first compares and analyzes the product leasing demand, installation and dismantling demand, and supply chain profits under both centralized and decentralized decision-making scenarios. Based on these analyses, a profit–cost sharing joint contract is designed to coordinate the supply chain. Furthermore, the interrelationships among key parameters are examined through a sensitivity analysis and numerical simulation. The results reveal that enhancing leasing information services increases both the demand for construction machinery and the platform’s operating costs. These costs are positively correlated with the product’s selling price, leading to higher purchasing costs for lessees. Similarly, improving information services for installation and dismantling raises the platform’s operating costs and enhances service levels, which in turn increases installation and dismantling costs for lessees. The findings demonstrate that within a certain range of cost-sharing and leasing-sharing proportional coefficients, the joint contract enables the supply chain to achieve Pareto optimization. This approach simultaneously alleviates economic pressure on lessees, improves construction safety, and promotes the integration of installation and dismantling services.