A central challenge in non-Hermitian sensing is that spectral singularities simultaneously amplify both the signal and environmental noise. We address this predicament in a double-chain Hatano-Nelson model featuring unidirectional interlayer coupling. At the exceptional-deficiency (ED) limit, the system exhibits a macroscopically degenerate complex spectrum and a pronounced non-Hermitian skin effect (NHSE), yielding a sensitivity that scales exponentially with lattice size $N$ while remaining robust across a six-order-of-magnitude detuning range. By introducing diagonal spatial disorder, we demonstrate that the NHSE is progressively suppressed, whith eigenspace cosine similarity analysis quantifying a well-defined fault-tolerance threshold. To reconcile the sensitivity-noise trade-off, we delineate at-ED and off-ED operating regimes. While the at-ED configuration imposes fractional-order noise amplification [signal-to-noise ratio (SNR) $\propto {\delta }^{-1/2}$] that saturates at a suboptimal plateau, migrating to the off-ED regime eliminates this geometric singularity and restores a linear scaling law (SNR $\propto {\delta }^{-1}$), achieving an SNR enhancement of several orders of magnitude. Crucially, this improvement is achieved while fully preserving the exponential sensitivity scaling, albeit at a slightly reduced absolute sensitivity compared with the strict at-ED limit. Our findings establish the off-ED framework as a concrete paradigm for next-generation topological sensors that reconcile extreme sensitivity with robust noise immunity.

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• Quantum metrology
• Quantum sensing
• Exceptional points
• Non-Hermitian systems