CSS <style> is static. I don't think you can do that... you might have to look for a different approach.
You can try using CSS variables. For example, (the code below is not tested)
<template>
    <div class="class_name" :style="{'--bkgImage': 'url(' + project.background + ')', '--bkgImageMobile': 'url(' + project.backgroundRetina + ')'}">
    </div>
</template>
<style>
    .class_name{
        background-image: var(--bkgImage);
    }
    @media all and (-webkit-min-device-pixel-ratio : 1.5),
        all and (-o-min-device-pixel-ratio: 3/2),
        all and (min--moz-device-pixel-ratio: 1.5),
        all and (min-device-pixel-ratio: 1.5) {
            .class_name {
                background-image: var(--bkgImageMobile);
            }
        }
</style>
Note: Only the latest browsers support CSS variables.
If you still see any issues with the :style in the template then try this,
<div :style="'--bkgImage: url(' + project.background + '); --bkgImageMobile: url(' + project.backgroundRetina + ')'">
</div>
 
     
     
     
     
     
     
     
     
     
     
    