That maxim is one that should be taken with a grain of salt. The culture of early A.D. Judea was very different from our own. In his first coming, Jesus could only live so much of human experience. Truth, on the other hand, is eternal. We can always look to His teachings to guide our decisions.
Imagining what Jesus might do is not always a good guide of actions. We can imagine anything. Truth and morality are more concrete. It is far better to look at Jesus' teachings and think how we might apply them to a situation than to imagine what Jesus might do.
The questions you, Monsieur Wellpride, should be asking is not, "Should Christians vote?" but rather, "Why do I not want to vote?" If you are afraid that your political opinions might offend someone, you would do best to remember this.
When Jesus was on this earth, He said things that the established spiritual and political leaders found very offensive. He did not come to placate everyone and bring them together. In fact, he made some rather harsh divisions. Read the gospels again, particularly this verse.
26 “If anyone comes to me and does not hate father and mother, wife and children, brothers and sisters—yes, even their own life—such a person cannot be my disciple.
The word "hate" in this context means " to set aside, to fail". Think of a teacher failing a student. They are saying that the student is inadequate to pass the course. If we are to follow God entirely, human relationships must come second to doing as He commands.