I don't know what two week GS you are thinking of. If you are getting an ATP with a type rating at CAE you will have to complete one of CAE's FAA approved training programs for that aircraft. I've worked for one of CAE's competitors, they had "short" courses for pilots upgrading or current on the aircraft. Maybe 24 classroom hours and 2 or 3 sim sessions and the checkride. I have no idea how CAE does it. Before you start that program you will have to complete an ATP CTP that CAE can provide. It's 30 classroom hours and 10 hours in a sim. I would hope if you are getting a type CAE would put you in the same type aircraft for the 10 hours CTP sim time but there is no connection between the CTP and your type rating training so they may not.
Your CAA has to verify your license and medical to the FAA and you have to say which FAA office you want the verification to go through. You should talk to CAE about it and do what they say, there are 3 FAA offices in the DFW area. And CAE will need the TSA background check so you should talk to CAE about it. You don't need a medical to fly a sim. But you are getting an initial issue. But you've validated your foreign medical. I don't know what CAE will want. You will have to meet their criteria for their training program since they are issuing the certificate. They may say you are good, they may say you need a third class medical.
The 142 school I worked for had a person who's full time job was processing TSA and FAA paperwork. I would think CAE would have a person doing the same job. You should talk to them.