Remove the ability to access it with a generic username and password. There is no way to identify who is deleting it if you allow that.
In other words, disable the shared mailbox AD account.
Then, delegate access ( full access and send as if needed) to the users who need access:
https://learn.microsoft.com/en-us/exchange/recipients/mailbox-permissions?view=exchserver-2019
Then ensure auditing is enabled for all actions for that mailbox.