Paramterized bulk copy - Disabling indexes for performance causes timeout exception

My requirement is to sling a rowset from one place in SQL server into a table in another place in the most performant way. I want this to be parameterizable -  I want to provide just a connection string and some SQL for the source and a connection string and a table name for the destination.  The package should do the rest. 

The solution I chose was an 2014 SSIS package with source and destination as ADO.NET connections configured from project variables.  The package has a script task to bulk copy the data.  For performance I disable the non-clustered indexes first.  But this preformance precaution causes the bulk copy to timeout after delivering the correct rowcount to the destination table.  Does anyone know what I can do to avoid this error?

Here's my script code:

                //get hold of the source and a data reader from it
                SqlConnection sqlconnSource = new SqlConnection();
                sqlconnSource = (SqlConnection)(Dts.Connections["source"].AcquireConnection(Dts.Transaction) as SqlConnection);
                SqlCommand sourcesqlCommand = new SqlCommand(SourceSQL, sqlconnSource);
                sourcesqlCommand.CommandTimeout = 1500;
                if (sqlconnSource.State==ConnectionState.Closed ) sqlconnSource.Open();
                SqlDataReader reader = sourcesqlCommand.ExecuteReader();



                //get hold of the destination and put the data in it
                SqlConnection sqlconnDestination = new SqlConnection();
                sqlconnDestination = (SqlConnection)(Dts.Connections["destination"].AcquireConnection(Dts.Transaction) as SqlConnection);
                SqlBulkCopy bulkCopy = new SqlBulkCopy(sqlconnDestination);
                bulkCopy.DestinationTableName = DestinationTable;
                bulkCopy.BulkCopyTimeout = 1500;
                bulkCopy.WriteToServer(reader);
                reader.Close();

This takes 128 seconds to put 13 million thin rows into my empty destination table and then throws an exception with this message:

 Timeout expired.  The timeout period elapsed prior to completion of the operation or the server is not responding. 

Any help gratefully received.