There is some new info I found out. Spec 3 had a physical release in the USA under the greatest hits version. There were two gh versions, and I don't know if they both have spec 3, but buy the gh disc that has the big red half oval on the disc bottom that is in red and says ps3 in it. That is spec 3.
Could the update be extracted from the disc and then applied to a Spec II install? Edit: probably yes, both discs have the BCUS-98158 model number
See if you can check the disc content, if it has a PKG in a folder called UPDATE or something like that, it's surely the Spec III update. It could also have the entire game packaged with all updates USA-PAL GT5Prologue is already on Spec II update
I can confirm the GH version like the one in the OP contains the spec III update. As a black label copy was already installed on the console, on the first boot of the red label version it asked me to install GT5 Prologue again, which could lead to two installs of the same game, but no. After answering yes to the prompt and restarting the game, the standard spec II install was upgraded to spec III. Ferrari California, Lotus Evora and GT by Citroën are on the dealership as they should and most important, the Ford GT is now controllable on High Speed Ring with professional physics thanks to "Improved responsiveness when using the wireless controller". Maybe the Eiger Nordwand time trials can be golded now. Save was carried over unchanged. What I can't confirm is if the update file is accessible through a PC. No one I know has a BD drive on their computer.