This paper investigates the boundedness and approximation properties of Hilbert-type singular integral operators within the framework of Triebel–Lizorkin spaces $F_{p,q}^s(\mathbb{R})$, a refined class of function spaces central to microlocal and harmonic analysis. We introduced a regularized Hilbert-type operator and established its strong convergence in Triebel–Lizorkin norms. Using Fourier analytic and interpolation techniques, we rigorously proved new boundedness results under optimal smoothness and integrability conditions. Furthermore, we demonstrated how this functional analytic framework enables robust modeling of non-smooth volatility structures in financial derivatives pricing, particularly under stochastic volatility regimes and market turbulence. This approach unifies singular operator theory and financial mathematics, offering a novel path for analyzing irregular price dynamics through spectral and geometric regularity tools.