onclick() and onblur() ordering issue

I was having the exact same issue as you, my UI is designed exactly as you describe. I solved the problem by simply replacing the onClick for the menu items with an onMouseDown. I did nothing else; no onMouseUp, no flags. This resolved the problem by letting the browser automatically re-order based on the priority of these event handlers, without any additional work from me.

Is there any reason why this wouldn't have also worked for you?

onClick should not be replaced with onMouseDown.

While this approach somewhat works, the two are fundamentally different events that have different expectations in the eyes of the user. Using onMouseDown instead of onClick will ruin the predictability of your software in this case. Thus, the two events are noninterchangeable.

To illustrate: when accidentally clicking on a button, users expect to be able to hold down the mouse click, drag the cursor outside of the element, and release the mouse button, ultimately resulting in no action. onClick does this. onMouseDown doesn't allow the user to hold the mouse down, and instead will immediately trigger an action, without any recourse for the user. onClick is the standard by which we expect to trigger actions on a computer.

In this situation, call event.preventDefault() on the onMouseDown event. onMouseDown will cause a blur event by default, and will not do so when preventDefault is called. Then, onClick will have a chance to be called. A blur event will still happen, only after onClick.

After all, the onClick event is a combination of onMouseDown and onMouseUp, if and only if they both occur within the same element.