Yes, you CAN do this.

Using pointer-events: none along with CSS conditional statements for IE11 (does not work in IE10 or below), you can get a cross browser compatible solution for this problem.

Using AlphaImageLoader, you can even put transparent .PNG/.GIFs in the overlay div and have clicks flow through to elements underneath.

CSS:

pointer-events: none;
background: url('your_transparent.png');

IE11 conditional:

filter:progid:DXImageTransform.Microsoft.AlphaImageLoader(src='your_transparent.png', sizingMethod='scale');
background: none !important;

Here is a basic example page with all the code.