What you are trying to achieve is not possible. Outlook templates do not store macros and while the Outlook editor is Word based it is only superficially like Word. If it was for your own use, you could add a macro to Outlook to enter or change the date in the message body, but you couldn't transport it with the template for other users.
If that will suffice then the following macro in your ThisOutlookSession folder will update and fix the date field (if present) when you click 'Send' for all messages.
Code:
Private Sub Application_ItemSend(ByVal Item As Object, Cancel As Boolean)
Dim olInsp As Inspector
Dim wdDoc As Object
Dim orng As Object
Dim ofld As Object
With Item
Set olInsp = .GetInspector
Set wdDoc = olInsp.WordEditor
Set orng = wdDoc.Range
For Each ofld In orng.Fields
If ofld.Type = 31 Then
ofld.Update
ofld.Unlink
End If
Next ofld
.Display
.Save
End With
lbl_Exit
Exit Sub
End Sub