Under the guidance of the dual-carbon strategic objectives, the continuous advancement of national energy conservation and emission reduction policies, coupled with the widespread adoption of green consumption concepts, the preference for low-carbon products of the consumers has become more popular. The manufacturing industry is accelerating its green transformation to deal with the above market changes. This study focuses on manufacturing enterprises with green transition intentions (green capacity demanders), examining two pathways to achieve green transition: independent development or acquiring green production capabilities through capacity-sharing platforms. This paper constructs differential game models for these two modes under distinct scenarios by establishing low-carbon goodwill levels as state variables. Specifically, when demand-side entities opt for platforms, decentralized and centralized decision-making frameworks are considered, and a cost-sharing coordination mechanism is designed. The findings reveal that low-carbon goodwill levels stabilize over time across all scenarios. Independent development does not always outperform platform-based approaches, as lower-cost methods exhibit superior performance. This mechanism effectively enhances low-carbon goodwill levels and overall supply chain profits, thereby achieving Pareto improvement.