You just need to override the same signature, then use your writer, then copy content from your writer to original writer:
Base Class Render (WebControl)
protected internal override void Render(HtmlTextWriter writer);
Override Class Render (My Classes)
public override void Render(HtmlTextWriter output)
{
// string builder is just an example of storage that custom TextWriter uses
var builder = new StringBuilder();
var myCustomWriter = new HtmlTextWriterExtension(builder);
base.Render(myCustomWriter);
myCustomWriter.Flush();
// write content written from base class to original writer
output.Write(builder.ToString());
}