How can I preserve soy sauce eggs?
You could ferment your eggs; salt-pickled fermented eggs are a thing and don't require any higher salt content than you already have.  However, there are some potential problems with that.

First, your comment that "a month later they've gone lacto on me. Probably safe, but I'm not a fan" gives the impression that you wouldn't like the result of fermentation.  Second, a couple of your ingredients -- the oyster sauce and the anchovies -- are easy-spoiling ingredients that could cause the batch to go off.

Your brine doesn't seem quite high enough in salt for canning them to succeed.  You can hot water bath brined eggs, but only if they are brined in a very high salt solution -- the linked post says 25% brine.  Heck, with enough salt you can keep eggs for months without canning at all.  But that's a lot more salt than you're using.

While you can theoretically pressure-can boiled eggs without salt or acid at all, after an hour of cooking under pressure they would not be edible.

Overall, I don't believe there's any way to preserve them that wouldn't substantially alter their flavor/texture.  So ... I'd say make as many as you'll enjoy in a few weeks.