I've looked at your document. If you can use macros, the following macro in your template should do the trick.
Code:
Sub AutoNew()
' See http://www.gmayor.com/installing_macro.htm
' Written by Charles Kenyon 2014-02-03
'
' update TOC fields in a document
Dim oField As Field
For Each oField In ActiveDocument.Fields
If oField.Type = wdFieldTOC Then oField.Update
Next oField
Set oField = Nothing
End Sub
Note, you will have to save this template as a .dotm template.