With such an early POD, the world would look very different, even if the Medes/Persians or someone else conquered the Neo-Babylonian empire at some later stage. Probably a somewhat earlier POD could ensure that the Medes/Persians were never able to rise as a significant power in the first place.
Part of the issue, is that it may be necessary for the Babylonians to become much more militaristic than otl. Assyria waged war on its frontiers for centuries in order to secure itself and successfully eliminated many foes for the Iraqi river valley urban culture.
Urartu: Near totally eliminated by Assyria, Cimmerians and Scythians (probably Medes also). However, the end of Urartu and the fall of Assyria is disastrous for Babylon as the former urban and farming society and kingdom of Urartu is now a staging ground for invasions from the north and east. Scythians, Medes and other assorted tribes roam the land formerly of Urartu and pose an extreme danger. Assyria previously dealt with this by invading and destroying these tribes before they could build up a strong enough warband to invade. Babylon likely will need to rely on the Tigris river to act as a wall against the hordes to the north; such s Wall is limited at best.
Cimmerians: The Cimmerians threat that previously plagued Urartu, from what I understand, was crushed by Assyria, and were sent into Anatolia. However, I imagine much of Cimmeria was integrated into the new Scythian hordes and forces that threatened Assyria.
Scythian: The fear from the north is nit to be dismissed and Scythia certainly could develop a force to invade Babylon and devastate it. Without Assyria, we lack that strong northern army that shields Iraq from the north.
Egypt: Doubtful to cause any real threat. However, they could be a venue of support for Scythian invaders or any other enemy of Babylon. Including the Chaldeans coming from below the Euphrates. If the Lakhmids are any example, an Arab for could be quite dangerous for the urban centers of Iraq.
Elam: Eliminated by Assyria, however it’s successor of Persia is immensely dangerous. Babylon must deal with them and constantly, preferably. Assyria’s policy of invading and dismantling the tribes of Iran was perhaps the only way in which the Iraqi civilization can defend its borders in a prolonged sense. This may require an extension across the Zagros and the creation of a standard army to defend the borders.
Medes: Similar threat to the Persian/Elam one. Crush them and dismantle their tribal structure, ally with internal tribes. Then perhaps create a system of settling tribes eastward to create more and more buffers. One issue to point out, it is perhaps inefficient to turn one horde against another. Say in the example of Scythian invasion of Media. The nature of these hordes is plunder and conquest, treaties will not be held for more than two rulers, thus any alliance with a major horde wound need to be discouraged in favor of attacking both whilst two engage one another, exactly as Assyria did.
Militarily: In terms of military, the most important point is that Babylon begins the creation of advanced forms of logistics to defend its fertile but deadly landscape.
Part of what Babylon needs a conception of or at least a stronger conception; is the concept that within their realm is prosperity, growth, civilization and nobility. The outside, in contrast, is one of inhospitability, barbarity and danger. Thus, it is is imperative for Babylon to extend its borders continually to create buffers and to maintain this. Assyria nearly had this attitude, however, it was more in terms of constant military invasions with an army, as opposed to creating strong borders and maintaining them as a civilizational demand and done with utmost fanaticism.
It does not seem outside reality that Babylon could conceive of this reality. They are a vast metrópole that will easily be able to se the exterior as barbarity.
So, I imagine the creation of a proto system of castles or fortresses being developed across the borders of the empire after these tribes have been dealt with with the Babylonian armies. These fortresses could be maintained with small numbers and could possibly neutralize the hordes to the north and east of maintained well. To maintain these, Babylon must develop some sort of logistical system to carry resources quickly, preferably across the the rivers and canals. These sorts of bureaucracy slums in my opinion best be controlled by the military at the start but over time, could be leased in control to merchants or nobles. The reason I favor this, is to give Power to third parties so to give positions to those who could rebel if not gifted. If an important position is given to a possibly rebellious entity, it could be beneficial in terms of them being covered in so much day to day work as to render their schemes less effective.
Further, Babylon should expand north into Dagestan over time. Slowly or rapidly pushing back the Scythians by allying with indigenous farmers. Push these tribes until you read the Caucus and then begin to establish a border zone. Set a standard army from southern Iraq in the mountains to maintain the creation of forts and the creation of gates at various passes that hordes pass through. The reason you choose southern Iraq, is to have soldiery without firm connections to indigenous peoples, so a rebellion is lessened. It is better to have an indigenous revolt amongst native peasantry and tribes than to have an alliance between one of your commanders and the indigenous peoples.
Regardless, Babylon by expanding this far north (at any cost over time) will decrease their expenses exponentially. By pushing north and placing forts across the region and gates, you decrease the number of soldiers needed to battle. Where less than 200 Babylonian warriors can defend a mountain pass against 10 000 nomadic horsemen, if they were required like the Assyrians, to continually campaign north, they must maintain an army exceeding 10 000 constantly and replenish their losses continually. The entire frontiers could be maintained with less than that and will acquire resources over time by trading with locals.
To the east, Babylon will do the same with the Zagros, create gates and forts in key locations. However, in the Zagros, migrating tribes east into Iran may be an option. Giving more foes for the Iranian tribes and then invading both intermittently with small forces of say 2k warriors to simply disrupt any unity and take resources back to the mountains where Babylon can horde horses, food, gold, etc... Zagros however, must be nearly impenetrable to assure Babylon’s survival. As it is the only natural defensive mountain to the immediate east and after it, is flat plains to be devastated. Babylon must expend any resource to keep this from occurring. Strongest and most elite soldiers must be placed in the Zagros and commanded to take strikes into the plateau often so to gain hatred for the Iranian tribes and to ensure that Iranians hate them. This will dissuade alliances between the two as was the case often in China. Babylon cannot afford a disastrous Manchurian invasion or fracturing as China did. Babylon’s population is much smaller and cannot sustain migrating peoples as well.
The southern east border, Elam, is another dangerous section. The Karun river is a necessity to defend and dominate totally. Expanding east and holding at least the most dense swamp regions is important. As the Zanj showed well, invasions from an Iranian Power can be utterly shamed by a small number of entrenched warriors in the wetlands or stretched across rivers and canals. Babylon should utilize this and create divisions of lightly armored warriors centered in the canals of Elam and the Karun, who can take the battle to the water and also on land against the dismounted Iranians. This could be done most beneficially with swimmers/fishermen from the land of Sumer who will be familiar with the ebb and flow of canals.
To the south, Arabia could be defended with minimal fortresses. The Arab tribes should be pitted against one another even if your client becomes dominant. Do not let a client become too powerful, if one does, turn Arab tribes against each other and or invade rapidly and with devastating force. Ensure that in such invasion, safe conduct is guaranteed to other tribes who side with Babylon. Other than this, do not allow large scale Arab migration to Iraq, limit this by extending small outposts southward to keep watch. When tribes move past an outpost, a Babylonian army is formed to capture said tribe and escort them to the Eastern or northern frontier to serve as shields against hordes.
Perhaps this is a start. Maybe I can expand more later.