Universal reencryption for signatures
If we have a signature scheme with deterministic verifier and a non-interactive protocol which is [co-non-malleable as an argument of knowledge] and statistically witness-indistinguishable, then we can get a fully non-interactive version of what you want by letting "mutate" be

"run that protocol's prover on", and letting "verify" be "run that protocol's verifier on".

If the protocol is only in the CRS model but has the property that

statistical witness-indistinguishability holds for all possible outputs of the CRS generator, then assuming the existence of a compatible private key would be enough leave the CRS model

by letting the "CRS" be part of the public key, since the signer would store enough about the generator's randomness to let one verify that the "CRS" is a possible output of the generator. ​ Section 8.3 gives such a protocol. ​ Given a statistically hiding commitment scheme,

this paper's proof of Theorem 4 lets "assuming the existence of a compatible private key"

be replaced with "letting the signing phase be interactive".