Wouldn't Dark Knight be more along the Chaotic Good side of the Alignment Chart?

Lawful Evil is usually reserved for those who works within their boundaries to exploit and twist into whatever they find most convenient. See every corrupt noble trope in the history of the planet.

The origin of Dark Knights in lore stems from a rebel Ishgardian knight who saw past the social dogmas and laws of the city to outright murder a high ranking priest because he was exploiting kids or something. That origin almost completely fits Chaotic Good to a tee with the whole rebellious and lack of concern over rules and laws in order to bring justice (good) on those who are clearly guilty.

FFXIV Dark Knights are still stalwart protectors, they fight to protect those who are too weak to protect themselves. It's just that a Dark Knight would probably just outright murder the super corrupt evil noble while the Paladin would argue that they should take him into custody and give him a proper trial.