Software-Defined Networking (SDN) separates the control plane and data plane of the network, abstracting the underlying infrastructure of applications and network services and handing it over to the SDN controller for unified management. The introduction of SDN technology into satellite networks effectively addresses control challenges posed by dynamic topologies and intermittent inter-satellite links. Due to factors such as geography and usage patterns, the distribution of remote sensing data traffic is uneven in both space and time. Some remote sensing satellites may acquire large volumes of data within a short period and transmit them back to the ground through intermediate nodes, leading to a sudden surge in load on certain controllers. This situation adversely impacts the performance of remote sensing data transmission. To mitigate this, switch migration operations are required to alleviate the load on overloaded controllers and to achieve the load balance among all controllers. In this paper, a switch migration scheme based on game theory is proposed under a three-layer satellite network architecture, which is designed based on the management costs and ensures timely triggering of the migration operations. Moreover, a game theory-based matching optimization model is formulated in the scheme to address the matching problem between target controllers and migration candidate switches. We further propose a distributed algorithm to seek the Nash equilibrium within the game. Experimental results demonstrate that, compared to other relevant algorithms, this approach can find the near-optimal matching strategy for each target controller and reduce the average latency of these controllers.