In this paper, we investigate a two-way unmanned aerial vehicle (UAV)-aided system, where two sources communicate with each other with the aid of an UAV relay. Two downlink transmission schemes consisting of non-orthogonal multiple access (NOMA) and network coding (NC) have been studied under finite blocklength regime. Considering imperfect successive interference cancellation (SIC), we have derived the exact and approximate expressions of the systems' average block-error rates (BLERs). In particular, we propose a solution to maximize system throughput by optimizing packet length. In addition, we have also quantified the effects of the system parameters on the BLER performance, including the transmit power, the power allocation coefficients, and the UAV's altitude. Numerical results demonstrate that the performance gap between the NOMA-based and NC-based systems is negligible.