Modify Notification Message When Profile Updated

You will have to use a gettext filter as there are no hooks or filters associated with your request.

function custom_user_message($translation, $text){
    if('Profile updated.' == $text){
        $current_user = wp_get_current_user();
        $foo_condition = ''; //Do some checking here for user custom field
        if(!$foo_condition)
            return $text.' Custom Message';
    }
    return $translation;
}
add_filter( 'gettext', 'custom_user_message', 10, 2 );