Translating lyrics for musicals presents unique challenges due to the need to ensure high translation quality while adhering to singability requirements such as length and rhyme. Existing song translation approaches often prioritize these singability constraints at the expense of translation quality, which is crucial for musicals. This paper aims to enhance translation quality while maintaining key singability features. Our method consists of three main components. First, we create a dataset to train reward models for the automatic evaluation of translation quality. Second, to enhance both singability and translation quality, we implement a two-stage training process with filtering techniques. Finally, we introduce an inference-time optimization framework for translating entire songs. Extensive experiments, including both automatic and human evaluations, demonstrate significant improvements over baseline methods and validate the effectiveness of each component in our approach.