In Summary point 7 is the direct answer to your question.
We don't have a huge amount of control over how the CSV renderer works within reporting services (without diving deep into RDL language). When I have a similar business requirement I; -
1) Get the underlying query of the data source of the SSRS report,
2) create an SSIS project
3) Create a data flow within the SSIS project.
4) Create an OLE DB connection manager within this data flow with similar details to the data source connection details of your SSRS report.
5) Create a OLE DB Source component which references the OLE DB connection manager.
6) Insert the underlying query of the SSRS report into the OLE DB Source component the data flow,
7) Create a flat file connection manager where you can define in more detail how the CSV file is to be created (making sure the check box labelled "Column names in the first data row") is unchecked.
8) Create a flat file destination component which refers to be flat file connection manager you just created.
9) Drag the green arrow from the OLE DB Source component to the flat file destination.
I hope the above helps.