Dynamic route's controller method not firing
Well, I'm an idiot. The problem was nginx after all. There actually was a block to handle CSS/JS assets that was preventing the dynamic route from executing because of a lack of physical files. I had to place the following above that rule:

location ~ ^/sites/.*/files/background_image/ {
    try_files $uri @drupal;
}


Here's the offending rule:

location ~* ^.+\.(?:css|cur|js|jpe?g|gif|htc|ico|png|xml|otf|ttf|eot|woff|woff2|svg|svgz)$ {
    expires 30d;
    tcp_nodelay off;
    open_file_cache off;
    open_file_cache_valid 45s;
    open_file_cache_min_uses 2;
    open_file_cache_errors off;

    location ~* ^.+\.svgz$ {
        gzip off;
        add_header Content-Encoding gzip;
    }
}


I found the solution after searching for relevant issues on other modules that generate dynamic files, such as imagecache, which led me to the official nginx Drupal guide. Definitely a learning exercise.