I don't think you want a unique CSS class for this.
If, and that is a big if, I have understood you correctly, you want to have the same style and layout for your HTML page, but with a different background image for the main <div> element.
So basically your CSS file should be static and not change for each game you want to show, only the background image of the <div> should change.
Have you tried do do something like this?
<div class="game" style="background-image:url( {{background}} )">
<!--
</div>
[EDIT] For some odd reason I typed 'removed' instead of 'url'. Must be lack of coffee.