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Capital constraints hinder enterprises’ carbon reduction efforts and affect the sustainability of the supply chain. To alleviate this limitation, the core enterprise considers offering two financial-based carbon reduction incentive mechanisms: cost-sharing mechanism (CS) and preferential financing mechanism (PF). In a supply chain with the dual sensitivity of market demand to price and carbon reduction, we model each incentive mechanism, discussing their impact, value, and selection strategies. The results show that neither party under CS pursues an excessively high share ratio. Only a below-threshold sharing ratio can promote the supplier’s carbon reduction behavior and improve efficiency for both parties. Conversely, PF has a stable incentive effect on the supplier’s carbon reduction behavior and can effectively increase the retailer’s profits. However, a reasonable carbon reduction standard is needed to attract the supplier. In addition, as market demand becomes more sensitive to carbon reduction, the feasible range of CS narrows and that of PF expands. We compare players’ preferences of PF and CS and find a Pareto region in which all players prefer PF to CS. Finally, we test the robustness of our findings by an extending model. Our study provides guidance for supply chain decisions facing dual pressures of financial constraints and carbon reduction.