Biometric authentication systems pose privacy risks, as leaked templates such as iris or fingerprints can lead to security breaches. Fully Homomorphic Encryption (FHE) enables secure encrypted evaluation, but its deployment is hindered by large ciphertexts, high key overhead, and limited trust models. We propose the Bidirectional Transciphering Framework (BTF), combining FHE, transciphering, and a non-colluding trusted party to enable efficient and privacy-preserving biometric authentication. The key architectural innovation is the introduction of a trusted party that assists in evaluation and key management, along with a double encryption mechanism to preserve the FHE trust model, where client data remains private. BTF addresses three core deployment challenges: reducing the size of returned FHE ciphertexts, preventing clients from falsely reporting successful authentication, and enabling scalable, centralized FHE key management. We implement BTF using TFHE and the Trivium cipher, and evaluate it on iris-based biometric data. Our results show up to a 121$\times$ reduction in transmission size compared to standard FHE models, demonstrating practical scalability and deployment potential.