You are using SIM and not AIM. If you transitioned to AIM you would have a lot more control over what happens during checkout and that is probably why they are referring you to it.
After they finalize their payment on authorize.net, then they should be returned to the page specified as the x_receipt_link_url hidden form element. You can add the url to the checkout_success.php page to that field and it will clear the cart automatically.
From the authorize.net documentation:
x_receipt_link_url Value: The URL of the link or button that directs the customer back to
the merchant’s web site.
Notes: For this field to be accepted as valid by the payment gateway,
you must also configure the receipt link URL in the Merchant Interface.
If the receipt link method is LINK, the URL specified becomes the href
value of the hyperlinked text. If the receipt link method is GET or POST,
the URL becomes the action of the HTML form.