But the rulers post-Babylonian Empire did not patronize the Mesopotamian gods to the extant of those who came before, ergo the religion faded and the area became fertile ground for Judaism, Christianity, gnostic religions, and eventually Islam.
This...isn't necessarily true. The Seleucids and the Arsacids rather heavily patronized the worship of the Anunna gods, and there were a number of Arsacid client states in Mesopotamia (Hatra, Aššur, Adiabene, Osrone) that continued to practice that religion as the state cult. And even without patronage, the temples were fairly effective as economic and social entities in their own right and were quite effective as maintaining the religiosity of the population while they remained intact.
The main issue is that the religion relies on either the state or the temple to provide for organization and direction, and the moment the Sassanids got into power they started burning down all the temples
much more efficiently than the Romans in their own corruption-hindered efforts to stamp out the old polytheistic cults. Even then, the old Mesopotamian religion endured in rural areas well into the beginning of the Islamic period, and in places like Harran and Mardin into the 1000s and 1400-1800s respectively.
Mind, the overall sentiment of "The easiest way to preserve the religion is to keep the local elites invested in the religion" is very true. Ultimately there's only so far a polytheistic religion can get in the region after the advent and dominance of Sassanid-era Mazdayasna or Islam (though for different reasons. The Sassanids were worried about syncretism with other polytheistic religions violating their new religious orthodoxy and therefore the state's legitimacy, Islam is just incredibly hostile to polytheism). The Harranians got by on large part due to the dual traits of Harran being fairly important to the Islamic, Christian, and Jewish faiths as a residence of Abraham and home of Seth, and by the fact that the remnants of Mesopotamian polytheism outside Harran's surviving urban cult were sufficiently low-key that they could be dismissed as 'mere' superstitions...a view that Harranian scholars at the court of the Caliph seem to have encouraged.
I'd say the latest method would be to have one of the post-Imperial Assyrian states, particularly Aššur or Hatra, conquer the others prior to the Sassanid Revolution, and use that event to break away and take as much of southern Mesopotamia as practical. Or alternatively aid the Arsacids in putting down the Sassanids and earn a spot functionally equivalent to an eighth Great House.