The Fundamental Review of the Trading Book (FRTB) poses a significant challenge for exotic derivatives pricing, particularly for non-modelable risk factors (NMRF) where sparse market data leads to infinite audit bounds under classical Martingale Optimal Transport (MOT). We propose a unified Rough Martingale Optimal Transport (RMOT) framework that regularizes the transport plan with a rough volatility prior, yielding finite, explicit, and asymptotically tight extrapolation bounds. We establish an identifiability theorem for rough volatility parameters under sparse data, proving that 50 strikes are sufficient to estimate the Hurst exponent within $\pm 0.05$. For the multi-asset case, we prove that the correlation matrix is locally identifiable from marginal option surfaces provided the Hurst exponents are distinct. Model calibration on SPY and QQQ options (2019--2024) confirms that the optimal martingale measure exhibits stretched exponential tail decay ($\sim\exp(-k^{1-H})$), consistent with rough volatility asymptotics, whereas classical MOT yields trivial bounds. We validate the framework on live SPX/NDX data and scale it to $N = 30$ assets using a block-sparse optimization algorithm. Empirical results show that RMOT provides approximately \$880M in capital relief per \$1B exotic book compared to classical methods, while maintaining conservative coverage confirmed by 100-seed cross-validation. This constitutes a pricing framework designed to align with FRTB principles for NMRFs with explicit error quantification.