Direct position determination (DPD) outperforms traditional two-step methods in accuracy. However, its practical application is hindered by two primary challenges: (1) the necessity to transmit raw data to the fusion center, which imposes significant demands on bandwidth and hardware resources, and (2) the absence of a closed-form solution for DPD, necessitating exhaustive search techniques and resulting in high computational complexity. To overcome these challenges, we introduce a subspace reconstruction-based DPD approach designed for an unmanned aerial vehicle (UAV) with a mounted array. This method requires a data transmission amount equivalent to that of two-step localization, specifically angle-of-arrival (AOA) localization while achieving performance comparable to the existing DPD method based on subspace data fusion (SDF). Furthermore, we introduce a numerically convergent solution based on majorization-minimization (MM) that guarantees convergence to a stationary point, thereby significantly reducing computational complexity and grid quantization errors (GQE) by eliminating the exhaustive search process. We validate the proposed method through computer simulations and real-world measurements conducted with a rotary-wing UAV, demonstrating its effectiveness and advantages.