Where I’ve come in regards to the Pharisees is that they were willing to make violent concessions for the sake of a tenuous status quo. Passover was often a flash point of rebellious activity in Jerusalem (which is why Pilate is there in the first place; to keep Jewish people suppressed and to put down any riots or revolutions from would-be messiahs). Violence was not infrequent at the time. And every time there was violence, Rome would take away more freedoms from Jews.

So the Pharisees are put in a position to see Jesus as a potential catalyst for Roman violence. So they figure that if they help hand over another would-be messiah then they can have a quiet Passover. But this mentality winds up being a sort of Leopards-eating-faces situation because Rome destroys Jerusalem a few years later anyway (due to a would-be messiah—just one that the Pharisees thought might be the real deal this time).