<p>The proliferation of new space constellations has paved the way for a multi-layered satellite network architecture, integrating GEO, MEO, and LEO satellites to provide global, seamless connectivity and computing services. However, efficiently offloading computational tasks in this heterogeneous and dynamic environment is a significant challenge. Existing task offloading mechanisms often overlook the unique characteristics of satellite networks, particularly the severe hardware constraints of high-altitude MEO and GEO satellites and the network resource overhead incurred by routing. This paper proposes a novel multi-layer satellite task offloading framework (MSTOF) for software-defined networking (SDN)-enabled space networks. In our architecture, GEO satellite acts as global controller (GC), MEO satellites as local controllers (LCs), and LEO forwarding groups from the data plane. The primary contribution of the MSTOF is a Quality of Service (QoS)-aware, two-tiered task offloading model, integrated with SDN in a multi-tier satellite network environment. At the GC, the GEO controller performs global task offloading (GTO) for coarse-grained load balancing. At the LCs, MEOs perform local task offloading (LTO), solving a detailed optimization problem that jointly minimizes a latency and energy while satisfying stringent QoS constraints for task priority, latency, and path reliability. By incorporating the comprehensive cost of offloading, MSTOF establishes a robust framework that enables service differentiation and efficient resource management in multi-layered satellite networks. The simulation results indicate that the MSTOF framework outperforms existing approaches with respect to QoS satisfaction rate, latency, and load balancing.</p>