Integration Services :: Parameterized 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 performance precaution causes the bulk copy to timeout after delivering the correct rowcount to the destination table. 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;

[Code] ....

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.