This paper investigates the performance of an unmanned aerial vehicle (UAV)-assisted full-duplex (FD) relaying system with non-ideal transceiver hardware (NITH) and residual self-interference (RSI). In the proposed model, referred to as the UAV-NITH system, each terminal is equipped with multiple antennas. Spatial modulation (SM) is employed in the UAV-NITH system. Closed-form expressions for the outage probability (OP) and system throughput are derived for both transmit antenna selection (TAS) and non-TAS schemes. The analytical results are validated through Monte-Carlo simulations. The findings reveal that FD transmission significantly outperforms half-duplex (HD) in the low-to-moderate transmit power region, while TAS provides substantial gains, particularly under low impairment conditions. However, both RSI and NITH introduce error floors that limit performance at high transmit powers. Moreover, UAV mobility has a critical impact, as higher speeds drastically degrade reliability, while operation at millimeter-wave frequencies increases OP but can be mitigated by employing more antennas. Overall, the results highlight the advantages of FD relaying with TAS for UAV-assisted networks, while emphasizing the importance of interference suppression, hardware-aware design, and UAV speed control in future implementations.