Image filename lookups are configured to be lowercase, so you would need to change this to catbg4.png if you want the background image to be substituted for the text. Only stylesheet tags use camel case.
If you just want the background behind the standard text, don't give it a keyword name, just something like fr-bg.png.
You can get the distinct styling without making "special categories", by making a define for catGroup4 (as you had originally, but with some incorrect heading tags). Use this as your model:
PHP Code:
define ('CAT_BOX_HEADING_1','1|0||');//new list - no headings
PHP Code:
define ('CAT_BOX_HEADING_4','1|0||');//new list - no headings
Now you can add rules for
#catGroup4 {}, like
Code:
#catGroup4 li {background-image: url("../images/fr-bg.png");}
This category group will include the "Shop By FR Fabric" category, so both will have the distinctive background image, as will their subcats.