[BRAN]

Interesting, it has a wooden button?

As I remember brute (dosed) force was necesarry for the (dis/re-)assembly, as those handles slide on grooved rails (within the handle). The wooden one's were subpar, so I had to use tiny precision files to shape those for a perfect fit. Once it sits, it sits.
I could imagine taking it off again will harm the fitment and you have to use glue later on for a permanent fit.

To put it in a nutshell: Go the route with reinstalling that missing upper piece.
You might want to ask in the general segment, if someone (Woody?) could check that on some spare brake handles.