Apply custom CSS to the popout window that is upgrade-safe

I've applied some custom CSS to the normal chat box on a site, but the poput chat is pretty ugly, uses bad fonts and is hard to read. The styles for the text is pretty much browser default. The popout doesn't load the theme styles or anything it doesn't need (which is awesome - but it makes it harder to custom style).

In looking at the source code, I know I could edit the plugin CSS file to get the look I want, but that won't last through a plugin update.

Is there any upgrade safe method you can advise me on so that I can throw some CSS onto the popout chat.