The "different computer id" is linked with the sig.bin file that is present in the GC2 folder. Remove it and follow the procedure. GC2 will attempt to validate the game. When it fails, it will offer you a way to the the sig.bin file from e-mail exchange.
BTW, the correct procedure to install a game with SDC from another computer, is to create an archive with SDC, move the archive to your galing computer, then use SDC to restore the archive.
And instead of installing directly the game on your connected computer, you could have used the option to download and archive the game.
Since your gaming computer isn't connected to internet, you will need to use a lot the SDC archive feature since updates are all done with SDC and you won't find any availiable standalone patch.