In the context of the cap-and-trade policy, this study establishes two-echelon game theoretic models of a low-carbon supply chain consisting of a manufacturer and a retailer to examine the optimal financing strategies of the manufacturer: (i) green credit financing (GCF), (ii) green credit financing with partial prepayment, denoted as mixed financing (MF). Under each financing strategy, this study examines both the static equilibrium and dynamic evolution of low-carbon supply chain members. We further compare the average profit, average carbon emission reduction, and local stability of the dynamic systems under GCF and MF. The results show that under static conditions, compared with GCF, manufacturer’s choosing MF can bring higher profit to supply chain members. When the initial per-unit carbon emission of the product is low, consumer environmental awareness, the carbon trading price, and the green credit subsidy have a positive impact on carbon emission reduction. Under dynamic conditions, when strategy adjustment speed is low, under both financing strategies, the manufacturer’s dynamic strategies always converge to equilibrium wholesale price and equilibrium product low carbon level. As the strategy adjustment speed increases, the supply chain system enters a chaotic state in which the supply chain members’ strategies deviate from equilibrium. By comparing the system stability and average profit under different financing mechanisms, the results show that choosing MF can damage profit and weaken the stability of the supply chain system as the system evolves.