This paper proposes a unified analytical framework for reconfigurable intelligent surface (RIS)-assisted two-way full-duplex (TW-FD) communication systems in 6G Internet of Things (IoT) scenarios. The proposed framework specifically addresses RISs with N reflective elements, facilitating efficient bidirectional communication. A novel unified moment-based analytical approach is developed, accommodating diverse fading models including Rayleigh, Nakagami-n, Weibull, Nakagami-m, and κ-μ, thereby significantly enhancing the versatility and practicality for complex 6G IoT environments. To comprehensively validate the applicability of our analysis, both independently identically distributed (i.i.d.) and independently non-identically distributed (i.n.i.d.) fading channel scenarios are investigated. By employing the Edgeworth expansion method, we derive analytical expressions for the probability density function (PDF) and cumulative distribution function (CDF) of the end-to-end signal-to-interference-plus-noise ratio (SINR). Additionally, closed-form expressions for probability, average symbol error rate (SER) for various modulation schemes, and end-to-end ergodic rate are provided. Monte Carlo simulations demonstrate the accuracy and robustness of the theoretical models proposed. The results presented in this work not only contribute substantially to the analytical methodologies for RIS-assisted communication but also offer practical guidance for the design and optimization of future 6G IoT systems.