Actually I did find the sollution.
Instead of placing the widget inside the container you want to convert to a drop down you need to place it underneath the container. See below:
The DropDownDivConverter is modifying the DOM. This causes some widget lifecycle functions to misbehave because the widget (in this case one of the buttons) tries to remove itself from the original parent node which no longer is the actual parent node.
Since your container is now the parent node, your solution is working fine in this case.
However I'd recommend to be carefull with this widget. I also encountered other inconsisten behaviour on context updates (i.e. a button in the dropdown would not pass the current context object, but an older one).