After working for a couple days with the BigCommerce team, I was finally able to get this issue resolved for my own use, so I thought the solution would be useful here for fellow Google travellers:
include_once './vendor/autoload.php';
use Bigcommerce\Api\Client as Bigcommerce;
Bigcommerce::configure(array(
'store_url' => 'YOUR_URL',
'username' => 'YOUR_USERNAME',
'api_key' => 'YOUR_API_KEY'
));
Bigcommerce::setCipher('RC4-SHA');
Bigcommerce::verifyPeer(false);
$ping = Bigcommerce::getTime();
if ($ping) echo $ping->format('H:i:s');
$orderID = "78165";
$options = Bigcommerce::getCollection('/orders/' . $orderID . '/products/', 'OrderProduct');
print_r($options);