LG,
First of all Gulfstream does not run their own factory school. It is actually Flight Safety that has the facility in Savannah.
Iver the years of going to FSI and Simuflite, I have meet many pilots that were doing their ATP in conjunction with their type rating. The Practical Test Standards (PTS) are the same for the type rating and the ATP. However, it appears that you do not have a FAA Commercial. That being the case, I think you will have to check the FAA website for the conversion requirements. If you know which facility you are going to, I suggest calling them as far in advance as you can and get a game plan lined up. This conversion could take some time.