"The Only Thing We Learn" by C. M. Kornbluth, first published in Startling Stories, July 1949, available at the Internet Archive.
•The story is set in a packed lecture hall
The professor, though he did not know the actor's phrase for it, was counting the house—peering through a spyhole in the door through which he would in a moment appear before the class. Tier after tier of young people, ready with notebooks and styli, chatting tentatively, glancing at the door against which his nose was flattened, waiting for the pleasant interlude known as "Archaeo-Literature 203" to begin.
•The lecture is about an ancient poem that commemorates the battle where the current empire captured Earth from the last empire.
"We shall examine first, by our archaeo-literary technique, the second book of the Chant of Remd. As the selected youth of the Empire, you know much about it, of course—much that is false, some that is true and a great deal that is irrelevant. You know that Book One hurls us into the middle of things, aboard ship with Algan and his great captain, Remd, on their way home from the triumph over a Home Suns stronghold, the planet Telse. We watch Remd on his diversionary action that split the Ten Suns Fleet into two halves. But before we see the destruction of those halves by the Horde of Algan, we are told in Book Two of the battle for Telse."
•The lecture is interspersed with flashbacks from the invasion under discussion.
More like one long flashback, from the point of view of the defenders of the empire under attack by barbarians of the Frontier League:
Wing Commander Artis heard the clear jangle of the radar net alarm as he was dreaming about a fish. Struggling out of his too-deep, too-soft bed, he stepped into a purple singlet, buckled on his Sam Browne belt with its holstered .45 automatic pistol and tried to read the radar screen. Whatever had set it off was either too small or too distant to register on the five-inch C.R.T.
•In the flashbacks a professor of history discusses the rise and fall of the Roman Empire.
Not the professor, but an archivist in the flashback alludes briefly to ancient Rome:
"I know of no organization called the Frontier League," Arris said. "If you are referring to the brigands who have recently been operating in Galactic East, you could at least call them by their proper names." Really, he thought—civilians!
"So sorry. But the brigands should have the Regulus Cluster by now, shouldn't they?" he asked, insinuatingly.
This was serious—a grave breach of security. Arris turned to the little man.
"Mister, I have no authority to command you," he said measuredly. "Furthermore, I understand you are enjoying a temporary eminence in the non-service world which would make it very difficult for me to—ah—tangle with you. I shall therefore refer only to your altruism. How did you find out about the Regulus Cluster?
"Eloquent!" murmured the little man, smiling happily. "I got it from Rome."
"Arris searched his memory. "You mean Squadron Commander Romo broke security? I can't believe it!"
"No, commander. I mean Rome—a place—a time—a civilization. I got it also from Babylon, Assyria, the Mogul Raj—every one of them. You don't understand me, of course."
•The professor is excited to finish the lecture, but is interrupted right at the end and forced to read an announcement
"I have been asked to make two announcements. One, a bulletin from General Sleg's force. He reports that the so-called Outland Insurrection is being brought under control and that there is no cause for alarm. Two, the gentlemen who are members of the S.O.T.C. will please report to the armory at 1375 hours—whatever that may mean—for blaster inspection. The class is dismissed."