Fully homomorphic encryption suffers from a large expansion in the size of encrypted data, which makes FHE impractical for low-bandwidth networks. Fortunately, transciphering allows to circumvent this issue by involving a symmetric cryptosystem which does not carry the disadvantage of a large expansion factor, and maintains the ability to recover an FHE ciphertext with the cost of extra homomorphic computations on the receiver side. Recent works have started to investigate the efficiency of TFHE as the FHE layer in transciphering, combined with various symmetric schemes including a NIST finalist for lightweight cryptography, namely Grain128-AEAD. Yet, this has so far been done without taking advantage of TFHE functional bootstrapping abilities, that is, evaluating any discrete function “for free” within the bootstrapping operation. In this work, we thus investigate the use of TFHE functional bootstrapping for implementing Grain128-AEAD in a more efficient base (B>2) representation, rather than a binary one. This significantly reduces the overall number of necessary bootstrappings in a homomorphic run of the stream-cipher, for example reducing the number of bootstrappings required in the warm-up phase by a factor of ~3 when B=16.