...It's surprising to me that it's this difficult to find a battery... I'm not looking to enter my W113 in any concours, but I want as close to period correct as possible...
You are looking for a unicorn, that's why it's difficult. If you are not entering a concours, particularly an MBCA concours, there's no reason to sweat trying to find something that basically doesn't exist.
A "period correct" battery might be considered to be one of those old "tar top" batteries. You can even buy fake tar tops for modern batteries. There used to be a company in Hudson, OH that made "old" period correct batteries, but I think they are gone and replaced by Jim's Battery Manufacturing 800-426-7580. There's a vintage battery company out of Denmark called SEH, and you can find them here:
https://www.classic-batteries.com There was another company, also out of business, called Restoration Battery in Cincinnati...
It must not be a lucrative business making "old" batteries as it were.
Take the advice here, buy a Walmart or other plain battery, and tart it up with Varta or Bosch or Beru or any other kind of old German electrical stickers you can find or make.