My better half loves the show. As you've observed they don't use a specific jet but now that I think about it I've never noticed them use anything other than Gulfstreams. The two most recent episodes I've watched have featured a G-IV and a G550.
I also love the show! the newer season hasnt been shown where I live yet, and I do have probably a stupid question and not to do with this topic, but do the real BAU use private aircraft like in the TV series or do they fly commercially??