The Assyrians fell to the Medo-Babylonian forces in stages, with their last capital city being Harran. Harran was seized in 610BCE, 54 years before Nabonidus became king. Crown prince Ashur-uballit fled to Egypt, and attempted to retake Harran in 609BCE but was repulsed.
This is the end of the political data I can find. From the maps made of the Neo-Babylonian territory, it seems to be accepted that Harran became a part of the Babylonian empire, but various sources also call it a Median city. Nabonidus's hesitation to complete restoration works on the temple to Sin there certainly seems to imply that it was not considered a safe zone, although he was content to march beyond it to Cilicia to put down a rebellion there in the third year of his reign.
Are there any economic tablets from Harran inscribed as being written during the reigns of the Neo-Babylonian kings? Not the famous Harran Stele's of Nabonidus, but just simple clay tablets with nothing whatsoever to do with anything important. About land tax, or payments of barley, that happen to helpfully date themselves at the bottom as '4th year of Nebuchadnezzar', or Neriglissar or any other.
It should be a simple thing to find out, but most of these boring texts don't get uploaded online, possibly because they're so dull nobody translates them at all and they're just sitting in a pile.
If anyone happens to know, I'd be curious to get confirmation that post-Assyrian Harran was a Babylonian state, and not a Median one.
Absolutely Babylonian. As someone who spends a lot of time reading and thinking about evidence to determine the borders of the Median Empire, I'm genuninely baffled and curious about what you're reading to suggest that Harran was controlled by the Medes. I'm also a little confused about why you want rule out the Nabonidus stele here, as a monument built by a Neo-Babylonian king celebrating his accomplishments and building projects in the city is a pretty clear sign that he ruled there rather than some other regional power.
No administrative archives have been discovered in Harran. There are probably two pieces of explanation here. First, it was apparently allowed to languish and fall into serious decline for most of the Neo-Babylonian Period. That would explain why the Elhulul, the temple of Sin, had to be rebuilt by Nabonidus (per his Stele) despite being a prominent administrative center in the Assyrian period. Second, the city's Great Mosque was either built over the Elhulul or used parts of the temple including old stelae and clay documents to fill the mosque's foundations in the early Islamic period. We know this because that's where archaeologists have found the few Babylonian texts from Harran, including Nabonidus' stele. That also includes three heavily damaged fragments of Neo-Babylonian dedication tablets discovered in 1959, discussed more here. They are almost unreadable, but appear to be short votive messages similar to sections of the Stele I've discussed so far. The tablet labeled "Fragment 2" also appears to be marked with Nabonidus's patronym, providing a clearer date.
There are also two additional Nabonidus-era stelae from Harran, making three total. Of these, the most famous one (the one called The Harran Stele by Wikipedia) was discovered in 1956 and labeled H1B by its primary translator, Dr. C. J. Gadd. In his translation project, Gadd identified another, more damaged stele discovered in 1906 as H1A. That one contains a section describing Harran/the Elhulul's recent history that records time based on the previous rulers of Harran, including both Assyrian and Babylonian kings with no Median intermediary. It reads:
During the time from Ashurbanipal, the king of Assyria, in whose rule I was born-(to wit): 21 years, 4 under Ashurbanipal, 4 years under Ashur-etillu-ilani, his son, 21 years under Nabupolassar, 43 years under Nebuchadnezzar, 2 years under Ewil-Merodach, 4 years under Neriglissar, in summa 95 years, [the god was away] till Sin, the king of the gods, remembered the temple... of his great god-head, his clouded face' shone up, and he listened to my prayers, forgot the angry command which he had given, and decided to return t]o the temple.
The third stele was also found in the 1956 excavations, labeled H2A by Gadd. It is not a direct product of Nabonidus, but a monument dedicated either by his mother, Adad-Guppi, or in honor of her death. It contains a nearly identical section to the one quoted above, documenting the idol of Sin's absence from the temple for 95 years. A second passage containing most of the same chronology also refers to how Adad-Guppi served all of the Neo-Babylonian kings. She was the high priestess of the Elhulul, and thus lived and worked in Harran. To serve the kings of Babylon, her city would be in their territory. Likewise, there is no mention of Median rule.
To view more of Harran's Neo-Babylonian history, we have to look outside of the city itself to other Neo-Babylonian documents that mention it. Harran was just far enough away from Babylon that it did not factor into many of the administrative records of Babylonian merchants and bureaucrats, but it does appear in the "Fall of Nineveh Chronicle," the primary historical document for the last days of the Assyrian Empire. The same document also contains an important section for answering your question:
From the month Simanu until the month Âbu - for three months - they subjected the city to a heavy siege. On the [lacuna] day of the month Âbu they inflicted a major defeat upon a great people. At that time Sin-šar-iškun, king of Assyria, died. They carried off the vast booty of the city and the temple and turned the city into a ruin heap The [lacuna] of Assyria escaped from the enemy and, to safe his life, seized the feet of the king of Akkad.
On the twentieth day of the month Ulûlu, Cyaxares and his army went home. After he had gone, the king of Akkad dispatched his army and they marched to Nasibina. Plunder and exiles [lacuna] and they brought the people of Rusapu to the king of Akkad at Nineveh. On the [lacuna] of the month [lacuna] Aššur-uballit ascended to the throne in Harran to rule Assyria.
So here we have documentation from the Babylonians that the bulk of the Median force left the campaign after participating in the sack of Nineveh in 612 BCE. The Chronicle documents more than a year from late 612 to early 610 where the Babylonians campaigned alone before the Medes sent a small force under one of Cyaxares' suboordinates to assist with the siege of Harran, described here:
The sixteenth year [610-609 BCE]. In the month Ajaru the king of Akkad mustered his army and marched to Assyria. From the month Simanu until the month Arahsamna he marched about victoriously in Assyria. In the month Arahsamna the Medes, who had come to the help of the king of Akkad, put their armies together and marched to Harran against Aššur-uballit, who had ascended the throne in Assyria. Fear of the enemy overcame Aššur-uballit and the army of Egypt that had come to help him, and they abandoned the city, and crossed the Euphrates. The king of Akkad reached Harran, fought a battle, and captured the city. He carried off the vast booty of the city and the temple. In the month Addaru the king of Akkad left his troops and their camp, and went home. The Medes, who had come to help the king of Akkad, withdrew.
So the Medes did participate at Harran, but left the city in Babylonian hands once it had fallen. Harran and Ashur-uballit appear for the last time in the Babylonian political record the next year:
The seventeenth year [609-608 BCE]. In the month Du'ûzu Aššur-uballit, king of Assyria, with a large army from Egypt crossed the river Euphrates and marched against Harran to conquer it. They captured [A town on the road to Harran]. They defeated the garrison which the king of Akkad had stationed inside. When they had defeated it they encamped against Harran. Until the month Ulûlu they did battle against the city but achieved nothing. The king of Akkad went to help, but his army but did not join battle. He went up to Izalla and the numerous cities in the mountains [lacuna] he set fire to their [lacuna]
At that time the army of [lacuna] march as far as the district of Urartu. In the land [lacuna] they plundered their [lacuna] The garrison which the king of [lacuna] had stationed in it set out. They went up to [lacuna]. The king of Akkad went home.
So here we have the final conflict over Harran. The Babylonians held the city against a final Assyrian attempt to reestablish their kingdom with Egyptian support and by the time Nabopolassar arrived himself, the situation was stable enough that he took the army and campaigned to the north instead. There are no Medes in sight, and the early passages in the Chronicle pretty clearly establishes that they were identified when they happened to be present.