In this paper, we propose a reliability optimization technique for low earth orbit (LEO) satellite systems that operate in highly varying temperatures. In the proposed technique, we harden the target satellite on-board computer considering two common reliability threats, soft-error and hard-error, at the same time. Based on an existing fault-tolerant scheduling, we show how the reliability and lifetime of the system can be quantified with respect to a given error handling policy. Then, using these models, we perform extensive reliability analyses and optimizations of the hardening degree for different functional safety requirements. We implemented the proposed technique in a widely adopted real-time operating system (RTOS), Real-Time Executive for Multiprocessor Systems (RTEMS), working on a real satellite on-board computer system, GR712RC. The proposed technique enables efficient explorations of the trade-off between the soft-error reliability (failure rate), hard-error reliability (expected lifetime), and other resource utilization indicators, like power consumption or CPU utilization. In particular, the proposed technique could be successfully utilized to co-optimize the lifetime and fault-tolerance concerning the unique ambient temperature profile of LEO satellites.