If the whole point is to save the "flavor" of the noble, and not to have a specific function, I'd have to say that you should really go back to what it originally was...
The Dungeon Master was a wierd guy that came to your fortress "dreaming of treasure" in nothing but cloaks and gloves (which I took to mean odd bondage gear... My first DM, I named "Lady Zon", because her first name was Zon, and I figured she'd be the sort of person who'd demand everyone address her as "Lady" at all times.)
The dungeon master makes the most sense if you are actually, literally, building some sort of dungeon. As in, you are going to protect your precious, precious gold by building a big maze with traps and beasties to protect it. For that reason, the notion of having to hit cavern, or possibly even some sort of mysterious structure or cavern ruins that will go into the game later, might be the best way to go.
If exotic animals are the key feature of such a class, it might be an appointable/automatically self-appointed noble that rises up when you have a sufficiently trained looney dwarf with a sufficient enough "dungeon" for them to master. If we are to keep the exotic animal type of key trait, it might mean domesticating enough sufficiently strange and monstrous (like giant lions and harpies or something) of a specific minimum number of non-mundane species and a large enough base menagerie, and enough animal-based kills to trigger Dungeon Master status.
Having a dungeon master might not give them, individually, specific new powers, but positions in DF tend to give you, the player, new ways to see into the workings of the fortress, and it might give players more control over the behavior of creatures in general. For example, while the Manager lets you assign jobs, and the Bookkeeper lets you view stocks, the Dungeon Master might give you specific breeding instructions, or a menu that gives you a better ability to train specific types of routines into your creatures, such as keeping "packs" of creatures together, or staying closer to your military dwarves, or the ability to reassign war dogs that you have assigned to dwarves previously.