In hybrid encryption we use an asymmetric encryption scheme and the recipients public key to encrypt a symmetric encryption key $K$. I will denote the resulting ciphertext $C_A$. We then use a symmetric encryption scheme to encrypt a message $M$. I will denote the ciphertext of the symmetric encryption scheme $C_S$. Finally we send to the recipient both $C_A$ and $C_S$. The recipient then decrypts $C_A$ to extract $K$ and then decrypts $C_S$ to get $M$.
I am aware that sign-then-encrypt is recommended for authenticating asymmetric encryption. However, in the case of hybrid encryption there seems to be two options, either sign $M$ or sign $K$. It appears to me both options could work provided the symmetric encryption scheme is authenticated, e.g., using an authenticated mode, encrypt-then-mac or similar.
My question is, what is the "best practice" to authenticate in such an encryption scheme? I am guessing there could be some more or less subtle reasons to chose one over the other.