The damping ratio is a crucial modal property that influences the sound duration and acoustic quality of traditional Korean bells. This study presents a high-precision measurement methodology utilizing non-contact acoustic resonance excitation and lightweight piezoelectric patches, specifically designed to address the limitations of conventional contact-based methods. Traditional accelerometers introduce significant errors due to mass loading and cable stiffness; in contrast, the proposed method employs a lightweight piezoelectric patch combined with selective acoustic resonance to minimize parasitic damping. Specimens with varying tin and silicon content were fabricated and tested. Verification results indicate that the coefficient of variation for the measured damping ratio decreased from 24% (observed with the conventional method) to less than 10%, demonstrating improved repeatability and precision. Notably, the damping ratio reaches a minimum at approximately 13 wt% tin, providing scientific validation for the potential optimized chemical composition of the Divine Bell of King Seongdeok. Furthermore, the damping ratio was observed to decrease as the resonance frequency increased from 110 to 245 Hz. This research establishes a technical foundation for accurate acoustic simulation and the modern restoration of traditional metallic musical instruments.