Under an online-offline dual-channel model,a three-tier supply chain network comprising manufacturers, retailers and consumer markets is investigated. A competitive equilibrium model among manufacturers is constructed to maximize profits by incorporating a carbon cap-and-trade mechanism, a funding budget constraint and a minimum carbon emission intensity requirement. To solve the Nash equilibrium problem, the Karush-Kuhn-Tucker conditions are employed to transform the NE problem into an equivalent mixed complementarity system under the convexity condition. Furthermore, MATLAB is used to perform sensitivity analyses on key parameters. As carbon trading prices rise, manufacturers must adopt adaptive management strategies to mitigate the shrinkage in production scale caused by increased carbon trading costs. As per-unit product carbon emissions rise, manufacturers must incorporate carbon reduction technologies into product R&D and manufacturing processes to reduce the risk of profit erosion due to increasing emission costs.With an increased capital budget, manufacturers are more likely to strengthen carbon reduction efforts, thereby avoiding the vicious cycle of expanding production first and addressing emissions later.