If you take a convergent encryption algorithm and replace the hash $H$ used to derive the key with HMAC-H, keyed with a secret key $k'$, does the resulting algorithm provide you with deterministic authenticated encryption?
By convergent encryption I mean content-hash keyed encryption, where the hash is encrypted with the user's key (e.g. like this). By deterministic authenticated encryption I mean the definition in the SIV paper by Rogaway and Shrimpton.
I.e. we have a convergent encryption algorithm: $C(k, m) = E(k,H(m))\ ||\ E(H(m),m)$.
We turn it into $D(k, k', m) = E(k, \operatorname{HMAC-H}(k', m))\ ||\ E(\operatorname{HMAC-H}(k', m), m)$.
Is $D$ a deterministic authenticated encryption algorithm, if you verify the HMAC?
Note: the encryption no longer converges for different $k'$, but that is fine. Only the properties of DAE are required here.
