Yeah, sorry. I've missed the part where you said you "bought the physical copy of AotI and installed it".
The patch is for the JP ver. of PSU:AotI, the one you can dl from the 1st post if you don't already have it physically (like most people).
That said, there's a problem with the patch on your computer : PsuIlluminusOff.exe shouldn't go away. I know what it do that ("move PsuIlluminusOff.exe BACKUP >nul"), but I don't exactly know why the cracked PsuIlluminusOff.exe from the patch isn't moved to the game root folder when other *.exe (like PSUC.exe or ClemLauncher.exe) are. To patch the game, I advise you to follow steps mentionned in a message above.
And finally, to play online on Clem', (edit : you may need) the Manual Patch 27.03.2017, depending if their launcher/server can update or not the client (my patch already include their 10.01.2017 patch).