I'm working through a similar problem, except I've been trying to split my fort into five separate districts. One at the surface, one at each cave, and one at magma. Each will be specialized to a specific industry, steadily increasing in value as you go deeper. So far, this is farming, then cloth/hospitality, then stone, then a bit of a catch all I'm still trying to distinguish built around the grand library, and of course metalworking. My ultimate idea is to have a society stratified into six levels: The most important people: the nobles, the too maimed to fight but not maimed enough to die war heroes, and the metal workers get burrowed at the bottom, the next most important people are cave 3 level, etc. Ultimately, I think roughly 30-50% of my population will end up being in the sixth class, the "untouchables", who for the purposes of fort lore will be the most hated and actively disrespected class, which consists of haulers. At least until I figure out minecart logistics. My factorio tendency to make every rail connected to everywhere else for future convenience may prove to be a hindrance, here.
Also for the purposes of fort lore, I'm not separating the forts completely, but instead centralizing and focusing production, and relying on the haulers for distribution. For example, *all* food is cooked on the farming floor, and then carried down. No self respecting metalworker would ever demean themselves by working something as menial and pedestrian as a stove, and No self respecting metalworker would allow someone willing to work with such shameful materials to live anywhere near them. The dichotomy being that in theory, every part of the fort should be in a constant power struggle in an effort to gain dominance (except the haulers, who just want somewhere they can sleep and eat without being subject to the lawlessness of the central staircase, which I think I will open to all the caves and the surface with no active defenses past the potential drop), while simultaneously completely relying on every other section of the fortress for it's very sustenance.
Also, all five main levels should have their own military squad, as well as a larger flex squad that moves everywhere. Ideally, haulers and citizens (one of the levels) should never be allowed to see each other, and haulers should not be allowed past the gates to any of the five citadels. Lore wise, I figure the citizens look at the haulers as subhuman beasts that can do basic tasks for food but should never be interacted with, so they leave out scraps in return for things being carried where they need to go.
So far I have the five citadels built, each with their own tavern, temple, and sleeping quarters, (Grand Library is on the 4th floor, and I don't really know how to split up books between libraries, and I have a *large* collection already anyway...) as well as a "main hall" of sorts. I also have maybe half my magma level populated. I don't think I split my fort fast enough, because right now everyone is friends and gets depressed as soon as they get burrowed...