In the low-carbon supply chain, when confronted with an external market offering higher profitability prospects, the monopoly manufacturer will prioritize serving that market, inevitably resulting in supply disruptions to local channels. This study delves into two distinct subsidy strategies employed by the government to avert supply disruption: a technology-driven "push" subsidy, devised to improve the ability to reduce emissions, and a price-responsive "pull" subsidy, geared towards influencing wholesale prices. Furthermore, if the external market spans international borders, the manufacturer must contend with the imposition of tariffs. Consequently, we analyze how these two subsidy types mitigate the risk of local supply disruptions within domestic and cross-border markets. The presence of carbon tariffs will have an impact on government subsidies and manufacturer capacity. Specifically, the surge in external market prices augments the strength of the effect. Alternatively, when the external market price is low, capacity has a negative impact on government subsidies. Moreover, we observed that when the manufacturer's capacity falls short, the government favors the implementation of pull subsidies. This arises from the fact that scarcity of goods tends to drive up prices, prompting the government to adopt the pull model to bolster consumption and alleviate the pressure on consumers.