Although many models exist to detect singing voice deepfakes (SingFake), how these models operate, particularly with instrumental accompaniment, is unclear. We investigate how instrumental music affects SingFake detection from two perspectives. To investigate the behavioral effect, we test different backbones, unpaired instrumental tracks, and frequency subbands. To analyze the representational effect, we probe how fine-tuning alters encoders' speech and music capabilities. Our results show that instrumental accompaniment acts mainly as data augmentation rather than providing intrinsic cues (e.g., rhythm or harmony). Furthermore, fine-tuning increases reliance on shallow speaker features while reducing sensitivity to content, paralinguistic, and semantic information. These insights clarify how models exploit vocal versus instrumental cues and can inform the design of more interpretable and robust SingFake detection systems.