The FAA does not have their guidance for a Part 142 training center available online. If you were doing the training under Part 61, it does look like you have to appear at a FAA Flight Standards District Office.
That's insane that CAE doesn't know what they can do. I've been working with a 142 school in Texas for about 4 years. (No, we don't do A330s) Someone calls up, they say they what type they want training in, we ask questions about their background and flying experience and we tell them they fit into whichever syllabus, it costs x amount, when do they want to start?
Have CAE call their FAA supervisor (POI) and find out if they can't read their own authorizations. I can't believe you are the first person who wants to do this with them. We get these type of questions from foreign students all the time.