Wi-Fi-based localization using Received Signal Strength (RSS) is widely adopted due to its cost-effectiveness and ubiquity. However, localization accuracy of RSS-based localization degrades due to random fluctuations from shadowing and multipath fading effects. Existing fading distributions like Rayleigh, $\kappa \! - \! \mu$, and $\alpha$-KMS struggle to capture all factors contributing to fading. In contrast, the $\alpha \! - \! \eta \! - \! \kappa \! - \! \mu$ distribution offers the most generalized coverage of fading in literature. However, as fading distributions become more generalized, their computational demands also increases. This results in a trade-off between localization accuracy and complexity, which is undesirable for real-time localization. In this work, we propose a novel localization strategy utilizing the $\alpha \! - \! \eta \! - \! \kappa \! - \! \mu$ distribution combined with a novel approximation method that significantly reduces computational overhead while maintaining accuracy. Our proposed strategy effectively mitigates the trade-off between localization accuracy and complexity, outperforming existing state-of-the-art (SOTA) localization techniques on simulated and real-world testbeds. The proposed strategy achieves accurate localization with a speedup of 280 times over non-approximated methods. We validate its feasibility for real-time tasks on low-compute edge device Raspberry Pi Zero W, where it demonstrates fast and accurate localization, making it suitable for real-time edge applications.