I don't want to frustrate you, quite the contrary, especially because you answered my own question :)
I don't think that a browser's contect menu is accessible via an ordinary script on a web page.
If what you are asking for was actually doable, then the browser makers would possibly consider this a bug and remove this behavior. Cross-browser, this behavior is very unlikely to be available today.
Why don't you capture mouse events, and whenever the mouse is directly in the area of the element below that you want to show the context menu for, push the covering element below, otherwise back on top?
That is one possiblity I could think of, basically revealing/exposing the hidden element depending on mouse position. Like cutting a hole into the overlay.
Or why don't you make the text field transparent and put the overlay below the text field entirely?
If this does not work out technically, then at least you have a point in filing bugs or enhancements against the targeted browsers.
BTW it looks like the context menu actually works if the user right-clicks directly at the position of the caret, so this might be another loophole for you to consider.