I am not sure that is possible, however the following may work. It saves the document, then saves it as RTF, copies the RTF file content to the clipboard, closes the RTF file and re-opens the DOCX file and deletes the temporary RTF file. Note that the code will have to be in the document template and not the document.
Sub SaveandCopyForm()Const strPath As String = "c:\path\"
Dim strFname As String
Dim oDoc As Document
strFname = "Filename"
Set oDoc = ActiveDocument
oDoc.SaveAs2 FileName:=strPath & strFname & ".docx", FileFormat:=wdFormatXMLDocument
oDoc.SaveAs2 FileName:=strPath & strFname & ".rtf", FileFormat:=wdFormatRTF
oDoc.Range.Copy
MsgBox "Document copied to clipboard"
oDoc.Close 0
Documents.Open FileName:=strPath & strFname & ".docx", AddtoRecentFiles:=False
Kill "c:\path\filename.rtf"
Set oDoc = Nothing
End Sub