How to use an image with Responsive Image style as a background?
What you're trying isn't going to work on a background image. You'd need to do this yourself with javascript. How i've done it previously is added data attributes to the div for each breakpoint e.g.

&lt;div data-mobile-image="mobilestylename/aaa.jpg" data-desktop-image="desktopstylename/aaa.jpg"&gt;


Then in javscript use matchmedia or a body class if its already being added by your theme to know which breakpoint you are in and then do 

var mobile_image = $('.mydiv').data('mobile-image');
$('.mydiv').css('background-image',mobile_image);