In this work, we propose a fiber Bragg grating (FBG)-based sensor for curvature measurements. Two gratings are inscribed through the protective coating in a specialty optical fiber using focused femtosecond laser pulses and point-by-point direct writing technology. One grating is inscribed on the central core adjacent to an air channel, while the other is inscribed on the eccentric core. The bending characteristics of the two-core fiber strongly depend on the bending direction due to the asymmetry of the fiber cores. A bending sensitivity of 58 $\mathrm{p}\mathrm{m}/{\mathrm{m}}^{-1}$ is achieved by the FBG in the eccentric fiber core over the curvature range of 0–50 ${\mathrm{m}}^{-1}$. Temperature and humidity cross-sensitivity could be significantly reduced by analyzing the differences in peak shifts between the two gratings. The sensor features a large sensing range and good robustness due to the presence of its protective buffer coating, which makes it a good candidate for curvature sensing in engineering fields.