If you are flying on United internationally but with a domestic connection, and if your connection is legal (not a stopover), then the SWU should apply to the entire trip. This is, of course, pending space availability. For instance, I went to Singapore with a stop in Chicago and another in Tokyo. The SWU was used the entire way. You might confuse things if you bring up the UDU to the agent on the phone. The SWU should take priority. I would call the 1k desk again and make sure you speak to an good agent. If the person gets confused, hang up and redial. Good luck and happy travels.