I'd like to chime in in support of Assyria. As I see it, Ancient Mesopotamia is the most oversimplified area-period of the mod. While there were four major powers in the region—Sumer, Akkad, Babylon, Assyria—only one (Babylon) or two (if you consider Akkad to be represented by early Babylon) are represented by a civ generally consisting of a single large city that occasionally conquers neighboring states (Shush, Jerusalem). The warfare and shifting geopolitics of Mesopotamia are basically absent.
Of course, the constraints on both area and time are extreme. I don't think Sumer can be done at all—it stopped existing as an independent entity around 2000 BCE, "only" a thousand years after the start of the game, and had previously been part of the Akkadian Empire for a while anyway. At most, an independent city could exist in southern Mesopotamia for Babylon/Akkad to conquer. Or, if the Babylonian capital is for some reason built at Ur or Uruk, the civ could get "Sumerian Empire" as its dynamic name.
On the other hand, I think Assyria has a convincing case. What I would do is spawn both Assyria and Babylon (representing Akkad early on) in 3000 BCE. The cities of Babylon and Assur would be very close—perhaps separated by a single tile—and that would make it fairly likely that one would conquer the other, representing the unification of Mesopotamia as your choice of the Akkadian, Neo-Babylonian, or Neo-Assyrian Empire. If one conquers the other early on, a respawn should be possible if the conqueror becomes unstable. Uneasy coexistence would also be a strong possibility. The warfare/instability would make both regions less likely to successfully defend against Persian, Greek or Roman conquerors (compared to a Babylon that currently has not much else to do than to fortify their single mega-city).
I don't think time is that much of a constraint if both civs spawn in 3000 BCE and have historical periods stretching to the 600s and 500s BCE. Space is the main issue. On the current map, Assur would most likely be 1N of Babylon, though you could also place it on the hill 1N of that (if not, that tile would be Nineveh). Babylon itself could be moved 1S, I think. The Euphrates could be diverted to keep flowing south of the city. This would make the area between the rivers oversized, but I think it's fine to enlarge important areas, especially if it's at the expense of the Arabian desert. A map change is not strictly necessary, however.
Gameplay-wise, Assyria would be fun and unique in itself—the mod lacks a pre-Classical civ focused on military conquest. It would also make the game of Babylon and basically every other surrounding civ more exciting. A human Babylon would probably want to conquer Assur as soon as possible, but building a strong enough military would come at the expense of more UHV-focused stuff, making for interesting tradeoffs.
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About other civ additions, I personally have a soft spot for Sweden, which would make both an oversized Viking civ and a Scandinavia full of independents less common. It would also have a different focus than the Vikings/Denmark-Norway, which are more about early Viking raids, invasions and travels, while Sweden would shine more in the Early Modern period. It should spawn in the 11th or 12th century.
A Central Asian civ is also a necessity, but my knowledge on the matter is not sufficient to comment further. (Not to mention that I don't think any convincing actually needs to be done.)
