Just how their name would be said if written in hangul. Ideal but unlikely: let the person choose themselves, with assistance. Much more likely but acceptable: some government clerk decides. The important/useful thing would be to have a hangul version set officially and not have multiple versions at the whims of bank clerks. I really meant canonical, I didn't say anything about accurate pronunciation. It goes like this for Koreans writing their names in English as well. Because there have been two official romanisation systems for kor-eng, you get a lot of people preferring one or the other, and then you get people who just prefer a certain English spelling.

edit: Just like in the opposite direction, English can't really properly represent Korean sounds. hangul is actually more versatile here because there are very few silent letters, etc, it's (with the usual exceptions) mostly spelling=sound.

Do you find that the katakana approximates the native pronunciation of your name well? The Korean government already has, if I recall, an official list of hangul readings for words/names in a lot of different languages. The thing that used to throw me off is French and Spanish words in Hangul that are common in English, like "genre" which is a loanword in Korean too but pronounced "??" (like.. "jang-leu"). To me, the korean version weirdly sounds more faithful (within confines of syllable blocks) to french "genre" than the english version does. Am terrible at french though.

Nope. That is a linguistic impossibility; mostly due to the lack of the schwa vowel (?) in Japanese (Korean does have that, so my name is easier in hangul as far as vowels are concerned).

The katakana do approach it tolerably, and it is in essence my name when speaking Japanese.

With katakana and hangul it is always a matter of compromising in one direction or another. If you have a very common name there will be common transcriptions that most people will stick to, but you can't prepare a list of all foreign names in the world, so inevitably two people will prefer different compromises, especially if they are fluent in that language.

Personally, I would be very much miffed if someone forces a specific katakana reading of my given and family name on me. That actually happened once when I was in Japan as an exchange student at Rikkyo University. I don't normally use my second given name, so on one of the numerous forms that inevitably end up in front of you in Japan someone added their own interpretation of how that would sound in katakana. Some college administrative worker took the katakana provided in earlier communication for my first given name and family name and made up their own attempt based on how my second given name might be read in English (I'm Dutch). I was not amused.

I did the same thing while living in Japan, but mostly because a rigorous, per-syllable transliteration led to a name too long to fit on most official forms, causing an occasional bureaucratic scramble.