Have you tried deleting (or moving away) the character config? The subdirectories in "My Documents\My Games\Final Fantasy XIV\user" contain the char config, GUI layout, macros etc. and I had an issue a year ago or so when the game seemed to hang after the char selection menu, while the screen was still black with the "Loading"-Symbol in the lower right corner being active, and after this my login worked again.

If you have not tried moving the contents of the user directory away, it may be worth a shot.