Could we use a more efficient hash for signature generation? Is it possible to use a more efficient hash - such as one based on GHASH - instead of a cryptographic hash for signature generation?

In other words, could a combination of a fast hash and the private key operation together provide the security that we expect from a cryptographic hash? Could such a scheme be proven to be secure assuming that the underlying primitives are secure?
As mentioned in the comments already, you do not need collision resistance. You can get away with target collision resistance (TCR). The security game for TCR considers families of hash functions and requires the adversary to select the message it will find a collision for before it learns under which function of the family it has to find it. This is applicable in hash-and-sign signature schemes where you have a fixed length signature scheme that works without first hashing the message (like RSA) and then add a hash function to handle arbitrary length messages by signing their hash (like RSA-FDH). With TCR, you select a random function index and sign it with the hash. 

One problem here, if you want to go for standard model security, is that the function indexes (keys) have to have length logarithmic in the message length. At least we do not know any domain extenders for TCR better than this. However, I will omit the details. A "heuristic" construction is the RMX construction by Halevi and Krawczyk.         

For Fiat-Shamir based constructions like (EC)DSA things are not that easy as you cannot just "additionally" sign the index. What you can do however is to reuse the random nonce (think of $g^r$ in Schnorr) as index as pointed out by Mironov  in this paper. Actually, that paper discusses a lot of the things I just wrote in more detail.

If you want to go weaker you start getting to run into issues. If you use the hash to compress the message you are signing, you need some sort of collision resistance (which TCR is). At least, you definitely need second-preimage resistance (which one might also consider a variant of collision resistance as you actually output a collision...). If your hash is not second-preimage resistant a malicious party, after seeing a message signature pair, can always come up with a colliding message for which the signature is valid.