Image styles with $settings['file_public_base_url']?
Finally got it working. It seems that currently there is a bug in the image core module that prevents the use of absolute paths in the $settings['file_public_path'].

For those interested, you can follow the issue where you can find the patch attached.

Note, that you must also set some rewrite rules on your static files server to get it working. I provided a docker example on how to configure this on GitHub, but mainly here are the main config:

Vhost configuration of the static file server

Configuration example provided for Nginx:

server {
    listen 80  default_server;
    root /files/domain.fr/public;

    index index.html index.htm;

    location @rewrite {
        rewrite ^/(.*)$ http://DRUPAL.SERVER.IP/files/domain.fr/public/$1;
    }

    location ~ ^/styles {
       try_files $uri @rewrite;
    }

    location ~ ^/core {
        rewrite ^/(.*)$ http://DRUPAL.SERVER.IP/$1;
    }
}


Drupal settings file

Add this to your settings.php file:

$settings['file_public_base_url'] = 'http://STATIC.WEB.SERVER';
$settings['file_public_path'] = '/files/domain.fr/public';