Soft robotic grippers demonstrate great potential for gently and safely handling objects; however, their full potential for executing precise and secure grasping has been limited by the lack of integrated sensors, leading to problems such as slippage and excessive force exertion. To address this challenge, we present a small and highly sensitive Fiber Bragg Grating-based force sensor designed for accurate contact force measurement. The flexible force sensor comprises a 3D-printed TPU casing with a small bump and uvula structure, a dual FBG array, and a protective tube. A series of tests have been conducted to evaluate the effectiveness of the proposed force sensor, including force calibration, repeatability test, hysteresis study, force measurement comparison, and temperature calibration and compensation tests. The results demonstrated good repeatability, with a force measurement range of 4.69 N, a high sensitivity of approximately 1169.04 pm/N, a root mean square error (RMSE) of 0.12 N, and a maximum hysteresis of 4.83%. When compared to a commercial load cell, the sensor showed a percentage error of 2.56% and an RMSE of 0.14 N. Besides, the proposed sensor validated its temperature compensation effectiveness, with a force RMSE of 0.01 N over a temperature change of 11 Celsius degree. The sensor was integrated with a soft grow-and-twine gripper to monitor interaction forces between different objects and the robotic gripper. Closed-loop force control was applied during automated pick-and-place tasks and significantly improved gripping stability, as demonstrated in tests. This force sensor can be used across manufacturing, agriculture, healthcare (like prosthetic hands), logistics, and packaging, to provide situation awareness and higher operational efficiency.