Your question is a very interesting one, but, while I do think that the part relating to the way in which the Roman state reconciled its role in the execution of Jesus is a perfectly sound question, the premise of the first part of the question isn't correct.

The reconciliation of Jesus' anti-imperial rhetoric

Like I mentioned above, I think the premise is incorrect: Jesus' teachings, were not particular full of anti-imperial rhetoric. In fact, famous lines like "Render unto Caesar the things that are Caesar's, and unto God the things that are God's" attributed to Jesus have been interpreted as specific instructions not to antagonize the Roman government (in that particular case, in responds to imperial taxes) and the line "My kingdom is not of this world. If my kingdom were of this world, my servants would have been fighting, that I might not be delivered over to the Jews. But as it is my kingdom is not from the world" which was allegedly uttered while he was being questioned by Pontius Pilate is often taken to mean that his teachings were separate from political activity.

I think you might be confusing the refusal of certain early Christian sects to worship the Emperor/Imperial Cult (which was certainly seen as an affront to the Roman state by many contemporaries and the main reason why so some early Christians were executed) with the preaching of Jesus himself.

The reconciliation of Roman culpability

This is indeed a fascinating question. The traditional Christian doctrine held that it had been "the Jews", not the Romans, who were to blame for the death of Jesus.This is not a medieval invention, but one which was already present in the earliest Gospels.

To quickly summarize the capture, trial and death of Jesus, most of the gospels write that Jesus was:

• betrayed by Judas, then;
• arrested by temple guards then;
• brought before the Jewish High priests (the Sanhedrin) who found him guilty, then;
• brought before Pontius Pilate, who (ascertaining that Jesus is not from Judea, but from Galilea) sends him to King Herod (the ruler of Galilea), then;
• brought before Herod, who mocks him but allegedly finds him to be innocent and sends him back to Pilate, then;
• brought before Pilate again, who then tells the Jewish elders that Jesus has done "nothing worthy of death" and (observing a Passover custom) then famously offered the people gathered outside to choose between freeing either Jesus or Barabbas (a rebel, not, as is popularly though, a murderer) and the crowd chose the latter, then;
• Flogged and executed by means of crucifixion.

Given this narrative, early Christian apologists had little trouble in blaming Jesus' execution on the Jews rather than the Roman state. The gospel of Matthew explicitly contains a passage in which the Roman magistrate Pontius Pilate literally washes his hands of the responsibility, saying to the crowd: "I am innocent of this man's blood. Attend to that yourselves.".

Now, scholars have pointed out that Pilates role in the trial of Jesus is an odd one and the Gospels seem to go out of their way to portray him (and by extension Roman authority) in a good (or at least ambivalent) light. He is made out as a figure who doubted the accusations brought against Jesus, who did not personally wanted to execute him and who in fact went out of his way to reason with the Jewish elders in favor of his release. This is strange, because Pilate, as the Roman magistrate, bore the ultimate legal responsibility for Jesus' execution. The right to sentence someone to death, was restricted to the Roman administration and he could have refused to do so. Furthermore, Jesus was executed by means of crucifixion; which was the Roman way of executing slaves and rebels. It was a very specific punishment and, according to the gospel of James, Pilate even added the inscription "Jesus the Nazarene, King of the Jews" to the cross on which Jesus was crucified.

So the likelihood that the Gospel writers deliberately downplayed the Roman part in Jesus' execution is generally taken for granted by most scholars. Apart from the textual hints pointing that way, the early Christians also had good political and religious reasons to lay the blame with the Jews/ a non-believing crowd, rather than the Roman state:

• "The Jews" (that is, Jews who did not consider Jesus te be the Messiah) were the principal religious adversaries of the earliest Christians (at the time best defined as, Jews who did consider Jesus te be the Messiah) before they established themselves more solidly as a separate sect/religion.
• Neither the Roman state nor Roman citizens would have been very receptive to a Christianity focused on Roman guilt. The Roman state, as shown later in history, could have been expected to respond quite hostile to such accusations; of which the Gospel writers/earliest Christians would have been well aware.

So to get back to your question,

How did the government reconcile the culpability of Rome regarding Jesus' crucifixion when Christianity was legalized and implemented by the Roman Empire in the 4th century

The answer is, that there really wasn't much that needed to be reconciled.

Some traditions did partly blame (or at least punish) the individual Romans involved, such as the Roman soldier who pierced Jesus with a spear while on the cross. In one version, this soldier (Longinus) was said to spend an eternity in a cave where a lion would attack him every night, only to wake up healed ... but, all things considered, 4th century Christian beliefs were already formed in such a way, that they did not blame the Roman Empire or the Roman Emperor for the death of Jesus; they blamed the unbelieving, Jewish crowd/priests and this version of events was happily accepted by Roman converts, to which it was probably tailored to begin with.

Adding to this, was the fact that early Christians tended to see their (or their ancestors) conversion to Christianity as being (spiritually) reborn and as such most Christians differentiated between pagan and Christian emperors; and, by extension, the pagan and Christian empires. In this view; Christ had been condemned by pagan Romans, who served a Roman Empire in which the pagan Gods were still dominant and these Romans and that empire, were not the same (spiritually or morally) as the Christian Roman Empire that followed it (with some hick ups) after the conversion of Constantine the Great and the various ecumenical councils held after 325CE under the auspices of the Roman state.

Suggested reading:

• On the Trial of Jesus, by Paul Winter
• The Historical Jesus Through Catholic and Jewish Eyes, by Leonard Greenspoon