Due to the rapid evolution of the space industry, the fusion of satellite and edge computing, known as satellite edge computing, has gained significant interest from both academia and industry. However, satellite edge computing always encounters challenges such as unreliable communication links and limited computational resources. To address these issues, we design an enhanced STAR-RIS Air-Space Integrated Network (SRASIN) with collaborative task offloading capabilities. Within this network, user devices utilize STAR-RIS reflection and transmission links to communicate with a UAV swarm and a Low Earth Orbit (LEO) satellite. Specifically, we design an iterative optimization STAR-RIS coefficient algorithm, incorporating penalties and linear search techniques, to maximize the data transmission rate of user devices. Subsequently, we introduce a multi-leader and multi-follower Stackelberg hierarchical game model based on potential games to model the negotiation process among user devices, the UAV swarm, and the LEO satellite. Through backward induction, we demonstrate that our game model can achieve Stackelberg Equilibrium. Finally, we design a Hierarchical Iterative Task Offloading (HITO) algorithm to optimize computing resource pricing, offloading modes, and offloading rates. Experimental results validate the effectiveness of our proposed model and algorithm in minimizing delay and energy consumption for user devices within the SRASIN.