.NET SQL Server Authentication Schema issue
I'm not sure what is preventing you assigning a default schema to the user. You can simply use:

ALTER USER [NT AUTHORITY\Authenticated Users] 
WITH DEFAULT_SCHEMA = [Production]


Obviously replacing the user/schema names as appropriate.

However, it would be best if you could update the code to use a schema prefix - some more reading on that is here (functional reasons) and here (performance reasons).