Dishonored Definitive Edition

I stumbled on this solution for the original Xbox 360 version:
Solution for Enough Coin to Disappear In Dishonored

It suggested that low chaos is not necessary. You could complete Witches of Brigmore in high chaos and as long as you finish with 10000+ coins you will still get this achievement.

I can certainly confirm that this worked for me on the Xbox One Definitive Edition. I killed just about every enemy, I got the high chaos ending, and the achievement popped.

At the start of the DLC I imported my low chaos save from The Knife of Dunwall because I finished it with just over 4000 coins.

There must have been some recent changes, because ManicMetalhead is the most recent to confirm that starting a new game with no previous save data, you will begin with 3200 coins and 6-7 runes.
Necrolampoon has confirmed that you can still get the achievement by beginning the DLC without importing a profile from KoD (in which case you begin with 3000 coins) and still killing everyone you see. JJ Nerdstrong has confirmed that if you start with a new game in this DLC, and DO NOT import a previous save, then when you collect your gear from the chest at the start you will be given 3000 coins.

From what I have read elsewhere there is about 7700 in the entire DLC. So as long as you explore almost every area you'll get over 10000 even without importing, but obviously having more to start with does make it slightly easier and reduces the margin for missing out.

BaldyGamer has also confirmed that you can earn the achievement by loading a high chaos save from KoD and finishing in high chaos.

For me this achievement popped along with the high chaos ending achievement (All Come To Ruin) during the credits.

The easiest way to do this without having to thoroughly search every area is to first do a Knife of Dunwall run on any chaos level while saving all of the coins and importing that save in the Brigmore Witches. If done correctly, you can finish The Knife of Dunwall with just shy of 7,500 coins with the remaining amount easy to acquire if you want to do this on a no kill/no alert playthrough. A breakdown of the total coins you can get is as follows:

The Knife of Dunwall
Starting Money: 2,000
A Captain of Industry: 1,553 (2,053 If you help Abigail Ames; If you hide bodies correctly can still be accomplished on a no kill play through)
Additionally, salvaging butcher equipment can net you between 200-260 coins
Eminent Domain: 3,169
The Surge: 0
Total: ~7,400

The Brigmore Witches
Choosing Your Mark: 500
A Stay of Execution for Lizzy: 1,655
The Dead Eels: 3,520
Delilah's Masterwork: 2,030
Total: ~7,700

Total: ~15,100

With this, you can buy a few favors or upgrades to help out your play style. This breakdown can give you an idea of if you need to search a level a little bit more or move on to the next.

For my play through, I did a high chaos run through the Knife of Dunwall killing anyone and anything and collected around 7,200 coins. I then ran through the Brigmore Witches dlc without killing anyone while still buying the boot upgrades to help with stealth to give me a low chaos ending. On the way I only grabbed money I ran into finishing with just shy of 11,000 coins.