It can be done if you:
1. Set up the drop down CC with different values for Display Name and Value
2. Bind that CC to an XML Mapping element by using the XML Mapping Pane
3. Make a selection with the drop down
When you want to 'fix' the CCs to show the values, run the
4. Run a macro to convert the CC type to a Plain Text CC.
Code:
Sub ChangeCCType()
Dim aCC As ContentControl
For Each aCC In Selection.Range.ContentControls
aCC.Type = wdContentControlText
Next
End Sub