Emission reduction in transportation requires advancements in aviation technologies enabling fully electrified propulsion, among which the use of superconducting technologies can provide untapped benefits given their low weight and minimal losses for high-power transmission. However, to enable their use in aircraft electrical power systems, additional considerations for electrical and thermal performance during faults is required in order to meet stringent aircraft safety requirements. In this work, a cryogenically cooled electric aircraft power system is studied under short-circuit conditions for different cooling media. This system consists of a fuel cell, high-temperature superconducting (HTS) transmission line, inverter, and motor, where each of their fault models have been explored. A trade-off study of the impedance of the fuel cell is conducted to identify the values at which the superconducting cable remains thermally stable after a short-circuit fault, as it is crucial that the cable does not experience a thermal runaway.