Extend WordPress REST API to show slugs for menu items

I had the same challenge, and fixed it by adding it to the response. Example:

function get_menu() {
    # Change 'menu' to your own navigation slug.
    $menu_items = wp_get_nav_menu_items('menu');
    foreach($menu_items as $menu_item) {
        // ALTERNATIVE: $slug = get_post_field( 'post_name', $menu_item->object_id );
        $slug = basename( get_permalink($menu_item->object_id) );
        $menu_item->slug = $slug;
    }
    return $menu_items;
}

add_action( 'rest_api_init', function () {
        register_rest_route( 'myroutes', '/menu', array(
        'methods' => 'GET',
        'callback' => 'get_menu',
    ) );
} );