I don't known if the code on the front changes with each trip, but it was 725 when I was on it earlier. I don't know if that helps.
LUL trains keep their number for the whole traffic day, unless they get reformed onto a different service. Specials will almost always run with the same number whenever they’re out, although there was some clarity needed over whether the train was to run as 725 or 746 this weekend for ~reasons.
Is there any way to track the progress of this train? I cannot find it on intertube
Unfortunately not today. Because of how Intertube/the TfL feeds work it depends on how the train is entered into the signalling system. A lot of the time specials run fully manually, avoiding them being unnecessarily regulated and allowing them to keep moving more easily if they end up ‘out of turn’ with scheduled service trains. It’s not fully insurmountable, but it means that if there are things to take control staff’s attention it expedites the smooth running.
This does also have the affect that it will not always appear on platform describes either, although arguably this isn’t a huge issue for an empty, non-stop service.