I think the bigger (more interesting) question really gets into what "language" is.

The first time I found out about registers was when studying Thai. It took me years to figure out that English has at least as many registers as Thai - and that I, perhaps more deliberately but no more skillfully than most, use them all depending on my audience. "Essay" is a funny way to look at it considering everyone from Shel Silverstein to William F. Buckley wrote "essays" and no two essayists write the same. So is a dialect a new language? When did Portuguese become Portuguese instead of the Portuguese dialect of Spanish? When will Brazilian cease to be Portuguese? When will Cuban cease to be Spanish? I would argue that this point is arbitrarily assigned.

At the end of the day, my question is, though the lens of "Usʒ", how do we use what letters we have to write down this phoneme?

At the end of the day, my point is, you don't. Fetch ain't gonna happen. English is an incredibly versatile language but the other words we have with a zh are either borrowed or ancient. "yoozh" is a new word with no alphabet and also

However most Spanish speakers can't hear the difference between /ʒ/ and /ʃ/ and they are not aware that vision /ˈvɪʒən/ and mission /ˈmɪʃən/ don't rhyme.

So there's that.

Look - goobster's articles indicate that people have been trying to make Fetch happen since 2009 or earlier but I'm with this guy:

Respelling systems deployed to show pronunciation in some monoglot English dictionaries (notably those published in the USA) represent ʒ as zh pretty much without exception. So one can say that writing zh is a well-established convention, despite the claim in Wikipedia that it is ‘ad hoc’.

The only European language that uses zh for ʒ (or for anything else) in standard orthography appears to be Albanian — not a language often learned by outsiders. I can’t think of any non-European languages that use it, either.

It is plainly, obviously yoozh to me and a yoozh is so obviously a horrible sound made by an orifice in your body you were previously unaware of that I'm revolted simply from typing it. The fact that there's argument about a fucking obvious combo like "zh" indicates that my revulsion is not unique.

Thus, the word will die, no matter how many writers want Fetch to happen.