I am not sure how familiar your educated Greek or Roman (unless they were living in Syria and Mesopotamia) were with the Neo-Assyrian and Neo-Babylonian empires.
They were familiar with Assyria and Babylon.
Herodotus wrote "Five hundred and twenty years the Assyrians ruled Upper Asia. The first to revolt from them were the Medes." He also stated "Assyria possesses a vast number of great cities, whereof the most renowned and strongest at this time was Babylon, where, after the fall of Nineveh, the seat of government had been removed."
Appian of Alexandria, reflected on the "rise and fall of cities, nations, and empires, about the fate of Troy, that once proud city, upon that of the Assyrians, the Medes, and the Persians, greatest of all, and later the splendid Macedonian empire."
The Bible of course mentions Assyria and Babylon so when the Greeks and Romans Christianized then educated Romans certainly would have been aware of those empires from that point on.
In the span of human history, extensive violence on minority groups doesn't make you evil it's the norm. To identify the "most evil" I think you have to look at other criteria.
I would argue for the ancient Assyrians or Hittites (or really most of the older ancient near-east kingdoms) because of their methods of execution. All nations have episodes of violence, murder, and cruelty in pursuit of wealth and expansion but the ancient near east kingdoms seem to have taken a huge amount of perverse joy in torture. Even the Romans preferred a quick strangulation but locking a man in a box so only his head protrudes and then painting him with honey to attract insects? Flaying people alive? Mass executions of the children of your enemies and making them watch? Stuff like that wasn't unusual for people who offended the ruling class in the ancient near east.
Yeah, Assyrian kings were particularly brutal.
"I built a pillar over against the city gate and I flayed all the chiefs who had revolted and I covered the pillar with their skins. Some I impaled upon the pillar on stakes and others I bound to stakes round the pillar. I cut the limbs off the officers who had rebelled. Many captives I burned with fire and many I took as living captives. From some I cut off their noses, their ears, and their fingers, of many I put out their eyes. I made one pillar of the living and another of heads and I bound their heads to tree trunks round about the city. Their young men and maidens I consumed with fire. The rest of their warriors I consumed with thirst in the desert of the Euphrates."
- Ashurnasirpal II