If you double click the dropdown button it is blocked by FBlockDropDown := True. The FBlockDropDown doesn't make sense. It is better to remove it and add ButtonClick(Sender) in ButtonDblClick(Sender: TObject) procedure. In that way the dropdown button works with every click. If you don't add ButtonClick(Sender), then the dropdown button does not make the SearchList visible with the 2nd click, when the SearchList is visible before the first click.
We'll investigate what is happening. The TTMSFNCEdit is used for other purposes as well so we need to be careful not introducing breaking changes there.