Also, not ideal but perhaps more workable for you particular situation, you can inject the CSS through a style class in the designer. Ultimately if your client is already digging through the custom theme files then they probably have designer access as well, and if they are determined enough could also change this, or even the component setup anyway, but it is an option. (subject to change at IA's whim)
Basically you can create a style class and then in one of the fields, say the background image field you surround your injection with a closing brace and end it with an opening brace.
There are many examples of this on the forum, here is one: