This study examines a cooperative non-orthogonal multiple access (NOMA) network utilizing an energy-constrained unmanned aerial vehicle (UAV) relay (UAVR) to expand coverage and improve network throughput. In order to provide energy to the UAVR, we consider the hybrid simultaneous wireless information and power transmission (SWIPT) method, which allows the UAVR to harvest energy from the source (i.e., sink node) signal. Herein, a hybrid time switching (TS)-based and power splitting (PS)-based relaying scheme is applied to improve the UAVR’s energy harvesting (EH) efficiency and the system performance. Given this context, we derive the closed-form expression of the outage probability (OP) for the sensors to evaluate the network performance. Based on the achieved analytical results, we apply the bat algorithm optimization (BAO) method to determine the optimal working point (as a fraction of received power and power allocation coefficients, and the 3-D positions of the UAVR) for the system such that the OP is minimized. The numerical analysis indicates that BAO is effective in both exploring and exploiting solutions, making it a suitable choice for similar non-convex optimization problems in continuous search spaces for cooperative NOMA IoT networks.