In this study, we investigate the use of simultaneous transmitting and reflecting reconfigurable intelligent surfaces (STAR-RISs) mounted on energy-efficient unmanned aerial vehicles (UAVs) to support satellite Internet of Things (IoT) communications served by low earth orbit (LEO) satellites. First, we propose a STAR-RIS-equipped UAV framework termed Integrated STAR-RIS and UAV (ISRU). Then, we aim to optimize energy efficiency by jointly adjusting the UAV’s flight path, STAR-RIS phase-shifts, and power allocation among IoT devices, all while maintaining equitable user fairness level. However, solving this problem presents considerable challenges due to the non-convexity and NP-hardness properties of the objective function and constraints. To address, our work introduces a Dinkelbach-based alternating optimization (AO) procedure termed Integrated Trajectory, Phase-Shift, and Power Allocation (ITPP). Our simulation results show that the integration of ISRU and ITPP can achieve 67% higher sum-rates than non-ISRU schemes and save up to 40% more energy than unoptimized trajectory schemes.