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Kōzuki is traditional or revised Hepburn I guess and the ō tells us that it's a long vowel.
In kana spelling however, ō --> ou (or oo depending on the kana) and ū --> uu.
I would personally go with Kouzuki as this method is the one that most accurately represents the way that the vowels are written in kana and this is how text is entered into a Japanese word processor using a keyboard with Roman characters.
Kōzuki and Kouzuki are the same, the only difference is that the first one is written using Hepburn and the other is written using "kana spelling". Kozuki on the other hand would be a different name. I don't think I've seen that spelling in anything official, but I might be very wrong here, feel free to point it out if I am. I guess an explanation as to why the Kozuki spelling surfaced could be that someone wanted to write her name on a picture or whatever and used the Hepburn version for reference but picked a font that didn't include "ō", thus ending up with a regular "o" instead.
I don't know, that's just a random guess.
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