Using a Different Windows Account for Invoke-Sqlcmd
You could grant the account that is going to run the job the necessary permissions onto the SQL Server to run the query.  That would be the easiest.  If you are going to use Task Scheduler, it even gives a method for specifying the account to use.

The next best would be to use SQL Authentication.

Finally, you can use PowerShell to invoke a new thread to run under a different windows account, that thread then to use Invoke-sql.  However, the password then needs to be stored somewhere somehow (usually encrypted in a file).  This is not the best solution and not one that I would recommend using.