Filter to change the content of 404 page

I have a 404 plugin that does basically what you’re needing (if I understand correctly), only it creates a new page (using the same templates from your theme) and registers it as the 404 page. If you already have an existing WordPress page (NOT php file) then you could use code similar to the following to turn it into your new 404 page. Note: You do need to customize this a bit. See the notes in the code below.

//redirect on 404
function redirect_404() {
    global $options, $wp_query;
    if ($wp_query->is_404) {
        $page_title = $this->options['404_page_title'];//replace with your page title
        $redirect_404_url = esc_url(get_permalink(get_page_by_title($page_title))); 
        wp_redirect( $redirect_404_url );
        exit();
    }
}

//Make sure proper 404 status code is returned
function is_page_function() {
    global $options;
    $page_title = $this->options['404_page_title'];//replace with your page title
    if (is_page($page_title)) {
        header("Status: 404 Not Found");
    }
    else {
        return;
    }
}

//Register Hooks
add_action( 'template_redirect', 'redirect_404');
add_action('template_redirect', 'is_page_function');

If you want to checkout the full plugin code you can do so here: http://wordpress.org/plugins/404-silent-salesman/

Hope that helps!