Using a hash of data instead of the full raw data for authenticated data in AES-GCM
If I understood you correctly, you will hash everything you need to authenticate and include that in the AAD. This will be secure, as long as the hash is a secure collision resistant hash function. However, I don't understand what you gain. You can compute on all of this data in the AAD but only send what is necessary for reconstructing it. In this way, you don't have to store/send anything more than needed. In addition, the GCM authenticator is actually much faster than SHA256/SHA512, so it will also be more efficient.