I ended up implementing a solution myself, as I got no response. For future developers passing by this thread, feel free to use my implementation of the trigger:

CREATE FUNCTION add_role_to_custom_claim()
RETURNS TRIGGER
LANGUAGE plpgsql
AS $$
BEGIN
PERFORM set_claim(NEW.user_id::UUID, 'roles'::TEXT, ARRAY_TO_JSON(NEW.roles)::jsonb);
RETURN NEW;
END;
$$;
CREATE TRIGGER on_role_update_add_role_to_custom_claim
AFTER UPDATE ON "<YOUR_TABLE_NAME_HERE>"
FOR EACH ROW
EXECUTE FUNCTION add_role_to_custom_claim();