How can I run a T-SQL Script using SQL Credentials?
As you describe it, it's not going to work.  The T-SQL script needs to run on a SQL Server which means that it needs to be run in an appropriately-authenticated session.

One option would be to write the TSQL script, and then provide it with a VBScript or CMD file or PowerShell script that'll launch the TSQL at the correct server with the right credentials.