You should be able to just apply the "Clear Cart" server behavior using the "Before page loads" trigger and move the resulting code to the bottom of the page manually. I've done that on your live site. Give it a try and let me know if you still have problems.
It may not work because cookies have to be cleared in the header. A work around might be to create a clear cart page and load it in a hidden iframe at the bottom of the checkout success page. If the current solution doesn't work I'll walk you through implementing the iframe solution.
Did this help? Tips are appreciated...
