I can't really picture any primal battles right now, and no matter how much I try to picture them normally I can't imagine them as being any less than massive, occurring in what would originally be the open world, likely making use of multiple parties.
For example, in my mind, the quest chain for Leviathan actually starts with sort of a dark memory of a past era, sort of a pursuing Umbral force that was locked away incidentally when we were, awaiting awakening into the Astral age. Its quest-chain contemplates that sleep, umbral forces as a general aspect, and eventually Leviathan as a concept, rather than a primal. From there, it likely goes into the war with the Empire, which then descends further into the idea of the primals as a driving power to Eorzea's magic and life-force, and why the Empire sees/saw this sort of chaotic yet personified heart as despicable. From there you start noticing strange things like prisoners and unaligned pirates alike being shunted into a forced military arm of the Lominsian navy, with Thaumaturges watching over the project, in seemingly simple preparation for an incoming naval battle. In reality it is anything but that. The battle itself is premise, and the new leader of the Archons (Louisoix still leader in name, but this guy, while undemanding and with a quiet sort of charisma, is leading a new agenda -- the usage of the primal's power by defeating and taking control of their physical forms) is preparing for Leviathen's capture. The training leading up to naval battle is grueling, effective, but more than practical knowledge it seeks to instill a mindset that can be used for Leviathan's summoning, with a third of the force just waiting to die, a third complacent veterans, the last third naive, believing they can take on anything -- between them the paradigmatic flow necessary to bring forth the Primal in the midst of bloody combat.
This battle probably won't be possible until having defeated Shiva, the other of the primals having to largely to do with the flow of time. Her quest chain results in the ability to use Echo upon oneself for two reasons: to see into the past, and to see what's happened in the past (effectively the present) while frozen in time, allowing one to be (in mind) in two places at once. That's what's used in this battle. People have to be present for the war in order to first gain access to Leviathan's domain. But the real fight comes with one's own ships, not in the original naval scrimmage, arriving just as Leviathan appears for the actual boss fight. This Archon business plays all sorts of fun on the NPCs you've been training with in the quest-line thus far. Once reached through the quest, Leviathan can be reached through finding the correct rites upon sailing to a correct location (I don't know what properties those follow just yet...). Naturally, no war is required beyond the first quest-chain. The instance will resize itself, and its loot, based on how many players they are, in a common Lominsian pirate-raid setup. It can take 16 or 24 or perhaps even 32 players. Perhaps even any non-rigid number between those (and of course with different breaks depending on the party size ARR uses).
Yet, for all that prelude, I can't really see the fight itself... Bummer.
Edit: I'd like to see some lesser boss battles of the Primals, though they'd likely be entirely separate. For instance, some key sea monster fights, one perhaps in little more than a dingy with a lot of trying to jump onto its shell, get to a fin and dig in or else jump back onto the little dingy when it starts to go underwater. Or perhaps one that involves more of a "fishing" aspect of giant proportions, fought on the beach.