Is Pilate's reluctance to punish Jesus in the Gospels a way of deflecting blame away from the Roman government and onto the Joos?
The Romans only used crucifixion for slaves and for crimes against the Roman state. If Jesus was crucified, it was because the Romans perceived him as an enemy of the state. The Sanhedrin would have had nothing to do with it, and no Jewish religious squabble would have had anything to do with it. All four Gospels explicitly state that the official reason for the crucifixion was that Jesus had claimed to be the King of the Jews (or at least that other people were claiming he was the King of the Jews). That was the placard allegedly placed on the cross, so if that's accurate then that was the crime.
It is also likely that the Temple incident played into it. Causing any kind of trouble at the Temple during Passover would get somebody killed immediately. During the Passover week, the Romans were greatly outnumbered by Jews coming in from the villages and they were paranoid about riots. Josephus describes multiple violent incidents that happened in the Temple courtyard, including riots and mass slaughters. So an assault on the Temple or a threat to destroy the Temple or a prophecy against the Temple could easily get somebody killed.
It seems that the trial before the Sanhedrin is a Markan fiction invented as a way to try to take blame for the crucifixion away from the Romans and put it on the Jews. The trial and the reluctance of Pilate are both highly implausible for a variety of reasons, and Jesus did not preach anything counter to Jewish law anyway.