I've never had a problem with mixed content from CSS. Where is your CSS file being served up from the UI or file system? As I mentioned I always use a relative url
without issue when using the CSS Stylesheets app in the UI. If the problem still exists, I wonder if your stylesheet is still cached and causing problems then.
Since you're using physical files and I'm assuming they are served up from that file location and not the CSS Stylesheet app, then as long as your stylesheet is in /App_Themes/Site/Style.css, the above should work as well. I've never come across a site where I needed to specify a domain in any of the css.