For anyone else trying to do the same thing as I was...
As well as adding the hook pointing to your FontAwesome CSS file, you'll need to update the CSS file itself to replace all references to '.fa-' with '.icon-fa-' since Wagtail prepends the classname you specify in the StreamField content type 'icon' attribute with 'icon-'.
You'll also need to override the font-family style rule in wagtailadmin/scss/core.scss by adding the following, either at the top of the FontAwesome CSS file itself, or in a seperate CSS file which you can also add to the hooks file:
.object .stream-menu button:before {
font-family: "FontAwesome" !important;
}
Hope this helps.