Being a TSA certified known shipper will help you get the cheapest international shipping rates because it opens many doors and avenues not available to those who aren't.
In response to the events of September 11, 2001, the U.S. Department of Homeland Security (DHS) instated new regulations in order to ensure the safety of domestic and international air flights. The TSA was made responsible for instating these new regulations and programs as well as overseeing all transportation by truck, rail, ocean, and air.
One of the programs the TSA instated is the Known Shipper Management System (KSMS). It is one of TSA’s most important implementations in order to strengthen air cargo security by establishing procedures for air carriers, indirect air carriers, and freight forwarders to distinguish between known and unknown shipping agents that deliver their cargoes for air transport.
What does this mean for securing the cheapest international shipping available? It means when you comply with these regulations, you'll have access to more opportunities that result in lower shipping rates AND better routes.