UI Click pass through

I can't think of a way to bubble the event further - so I'd be inclined to get creative...

You can fire your own ray through the screen, and assuming your UI is added to the appropriate layer - you can get a handle to all the UI elements you "touch".

Then handle each one independently. The only issue though is that this does not use the UI message system, so sounds clunky...

Why do you need a transparent panel to get clicks but not block them? That's kind of contraproductive. ;)
Wouldn't it be easier to simply check if the the mouse is clicked & the context menu is not clicked? Because this means logically, that anything in the "background" is clicked. Example code:

 if(click & click.tag != "ContextMenu")
 {
     CloseMenu();
 }

Using Tags is just one simple way to give you an example.

Additionally I's like to mention, that this is a "HowToDo" question for Unity Forums, not for Unity Answers.
Unity Answers is for problems with Unity functions that behave unexpected.
Check out the Frequently Asked Questions and the User Guide to avoid this in the future.

Is there a way to have a UI element get handled but not block other elements?

This would the question you should google or search Unity Forums for.
Google might point you to Unity Forums anyway :)

Have a nice day.