register_post_status invisible but searchable

You can filter your admin query with pre_get_posts filter.

function wpse_306361_admin_search($query) {
    if(is_admin() && $query->is_main_query() && $query->is_search()) {
        $query->set('post_status', array('publish', 'draft', 'what-you-want', 'your_custom_status'));
    }
    return query;
}
add_action('pre_get_posts', 'wpse_306361_admin_search', 10, 1);

It will check if query is in admin, if it’s main query and if it’s a search query.

If it’s only for a post_type, don’t forget to add a condition in your if statement.

pre_get_posts on codex : https://codex.wordpress.org/Plugin_API/Action_Reference/pre_get_posts