Hi,
If I understand properly, you are attempting a replication (from ___, external? Which provider?) to Cloud SQL and you cannot turn on binary logs with Google.
Since Cloud SQL is a managed service, you will be restricted from certain accesses. For example, superuser is disabled and to turn on binlogs, you do it according to these[1] instructions.
If you have further issues after this, please link us the official article of the replication process at the source database. Without that, it's hard to tell if we are following standard procedure. From our end, the official procedure looks like this[2], make sure you're not missing any official steps from both providers.