Rapid and accurate body weight estimation is critical in emergency medical care, as it directly influences treatment decisions, such as drug dosing, defibrillation energy selection, and fluid resuscitation. Traditional methods such as stand-on scales, length-based tapes, or transfer-based weighing scales are often impractical for immobilized patients, inaccurate, or labor-intensive and time-consuming. This paper introduces MelodyBedScale, a non-intrusive and rapid on-bed weight estimation system that leverages bed vibration induced by music. The core insight is that body weight affects the vibration transfer function of the bed-body system, which is captured using vibration sensors placed on opposite sides of the bed. First, we identify weight-sensitive frequency bands and compose clinically acceptable soft, natural music with high signal energy in these frequency bands. This music is then played through a speaker mounted on the bed to induce bed vibrations. Additionally, to efficiently capture the complex weight-vibration relationship with limited data and enhance generalizability to unseen individuals and weights, we theoretically analyze the weight-vibration relationship and integrate the results into the activation functions of the neural network for physics-informed weight regression. We evaluated MelodyBedScale on both wooden and steel beds across 11 participants, achieving a mean absolute error of up to 1.55 kg.