In that case you will need a macro. The following should do the job, copying the formatted text of the content control containing the cursor to the clipboard.
Code:
Sub CopyCC_Content()
Dim oCC As ContentControl
Dim oRng As Range
On Error GoTo lbl_Exit
For Each oCC In ActiveDocument.ContentControls
If Selection.InRange(oCC.Range) Then
Set oRng = oCC.Range
oCC.Delete
oRng.FormattedText.Copy
ActiveDocument.Undo 1
Exit For
End If
Next oCC
lbl_Exit:
Exit Sub
End Sub