Not Square-Enix support here, but... given that it's only one computer that has this problem, I wonder if something's corrupted in the local data for that character.

To test that theory, on the computer that's affected, you might want to try temporarily renaming the "{Documents}\My Games\FINAL FANTASY XIV - A Realm Reborn" directory to something else and then running the game. Downside is you'll lose your HUD customizations and hotbars (unless you have them stored on the servers to download again), but if it is the local character data then things should work to let you log in s King Undercroft.

(And if you can load in that way, you can always upload the hotbars, gearset, and HUD configuration from the working computer and download it on the non-working one.)