Yes, you can click through div to underlying elements by following the below mentioned way:-
You can do this by using the pointer-events: none along with CSS conditional statements for IE11 you can get a cross-browser compatible solution for this problem.
Another thing you can do is by using AlphaImageLoader, you can even put transparent.PNG/.GIFs in the overlay div and have clicks flow through to elements underneath.
See the CSS you can use below:-
Below is the IE11 conditional:-
background: none !important;