I think you need to use fab as the prefix. From what I know, the b in fab is for “brand”. So your icon has either fab for brand, fas for solid, or fa for the regular version.
So I think it’s fab-mastodon and fab-bluesky. And there is no version like fa-facebook. Maybe you prefer one of the other Facebook icons?
Also, don’t forget to add the icons’ names to the custom_font_awesome_icons setting.