Answer

My sins are what caused this remarkable a fortiori inference to escape my notice. It is indeed a challenge.
Because of the dear esteem of the master, I will wave over it my dull hand, like one waving a sickle over standing grain.
 
First possibility:
The a fortiori inference is not meant to examine the balance of power between empires. Rather, it is an inference about God's governance of the world—whom He would prefer more to destroy.
The Chaldeans, who destroyed the Temple built by the Jews, are certainly very deserving of destruction. He delivered them into the hands of the Persians, meaning that the Persians are His faithful and important agents for important missions. But the Romans destroyed the Second Temple, which had been built by the Persians, and that is less significant; so the Romans are apparently only His second-tier agents (He assigned them a relatively less important mission). If so, the Persians are more important in His eyes than the Romans. Therefore the Romans will fall into the hands of the Persians.
Of course, this still requires investigation as to when this would happen, since neither these nor those are in the world anymore, and the Romans did not fall into the hands of the Persians. It is an a fortiori inference refuted by reality. But if the purpose of the inference was only to show that it is important in the Holy One's eyes to destroy Rome, while the Sages thought the Persians would do it (because they were the likely candidates at the time), then part of it was fulfilled. Rome was indeed destroyed, just not by the Persians. God's governance did in fact bring about the destruction of Rome.
 
Second possibility:
The a fortiori inference really is about the balance of power between empires. The Chaldeans were very powerful, since they succeeded in destroying a Temple built by Jews. The Romans destroyed a Temple built by the Persians, which required less power. Therefore it is obvious that the Persians, who overcame the immensely powerful Chaldeans, would certainly overcome the less powerful Romans.
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Shua:
I too had my sins hide this a fortiori inference from me, and so to this day I did not understand what the translator wanted in Lamentations 4:22:
"Rejoice and be glad, daughter of Edom, who dwells in the land of Uz; to you too the cup shall pass—you shall become drunk and strip yourself bare"
"Rejoice and exult, Constantinople, city of wicked Edom, built in the land of Armenia, with many populations from the people of Edom; punishment too is destined to come upon you, and the Persians will hunt you down, and the cup of curse shall pass over you; you shall be drenched and laid bare"

And one can add a bit, that from the words "built by the descendants of Shem" it sounds as though the issue is the familial hierarchy between the builder and the destroyer. The Chaldeans are descendants of Ham or Japheth (see Nachmanides on Genesis 11:28. And here too the translation in Lamentations takes a side in chapter 5 verse 8: "Slaves ruled over us" – "The descendants of Ham, who were given as slaves to the descendants of Shem, ruled over us"); and the Romans are, as commonly accepted, descendants of Esau, from the descendants of Shem. And the a fortiori inference is like the second possibility: the Chaldeans were very powerful, for they skipped levels in the proper hierarchy and destroyed the descendants of Shem, whereas the Romans did not overcome all that much, since after all they destroyed those lower than themselves; therefore it is obvious that the Persians would overcome them.
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Rabbi:
It should be noted that the verse does not say that the Romans will fall into the hands of the Persians, only that they too will fall. That indeed happened. The interpretation of the Sages or the translator apparently was not fulfilled.
At first I did not understand how the translator writes that Constantinople is the city of Edom. I thought the Eastern Empire was later. But now I checked, and indeed it was founded already in the 4th century CE (and only its name in Israel, Byzantium, is later). Constantinople, named after Constantine, has existed since then. Interesting.
 
Now I saw on Wikipedia about the Book of Lamentations:
Likewise, "Edom" mentioned in the scroll (Lamentations 4:21–22), "Rejoice and be glad, daughter of Edom, who dwells in the land of Uz […] Your punishment has been reckoned, daughter of Edom," is interpreted by him as a designation for "Rome." In his commentary to these verses, he incidentally reveals a Byzantine period of composition.
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Shua:
I forgot the main point and managed to come up only with the secondary one (the translation in Lamentations). But here is the main point: [Years have passed since I read the wonderful book on a fortiori inference in the Hermeneutical Principles series] are there other examples of an a fortiori inference that teaches not a law (something a person or a frog has to observe) but a fact of reality, and is it reasonable to assume that the parameters of such an inference are identical to all the rules of the hermeneutical principle of a fortiori inference? (For example, here one is of course deriving punishment from an inference. But the question is, for example, whether the components of the rule of "it is enough" that require verses would also apply to such an inference.)
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Rabbi:
Many thanks.
In a factual a fortiori inference there would be no rule of "it is enough" at all. In reality, "it is enough" has no meaning (and perhaps neither does a fortiori inference itself). The rule of "it is enough" deals with what we know about reality, not with reality itself. So if I know that a lion is stronger than a leopard, and I know that a leopard can overcome a cat, then it follows a fortiori that a lion can overcome a cat and anything weaker than a leopard. That is what I know. But obviously in reality a lion can also overcome something stronger than a leopard; I just cannot know that from these data alone (that it overcomes a leopard).