The easiest solution I can think of is to use the custom group name if the translation can’t be found. The only problem I’m seeing is that group names seem to be converted to lowercase.
I18n.t("emoji_picker.twitch", { defaultValue: "twitch" })
The more involved solution would be to make SiteTextsController and TranslationOverride work with custom emoji group names (or any other custom translation that needs to be added at runtime).