A couple of things.
First, you can't use a windows authentication domain/login name in a custom
connection string to connect to a SQL in SPD. Windows Authentication is
turned off.
Second, The Data Source dialog in SPD has a textbox for SQL server name, not
a dropdown list. There is a dropdown for tables AFTER you have supplied the
servername. That dropdown won't populate unless you authenticate correctly.
You can't do it with Windows credentials unless you use SSO. For a
connection string you would need to use SQL credentials, which don't have a
domain name.