Should a judge do it's job or maximize overall happiness

This thought experiment is a little more detailed than it needs to be to ask the question you're interested in.

First, your post is overly complicated and without a summary upfront. This makes it hard to understand your point, whatever it might be. Second, consider running your top-level posts by a native English speaker before posting.

Now, Bob is clearly not very smart, having been convicted twice for the same offense and still keeping at it. Carol's actions are less-than-believable, as a billionaire would have resources and counsel to channel her lobbying efforts in a legal or at least a more covert manner. Furthermore, judges do not act as expert witnesses. Fortunately, it is immaterial whether John is a judge or an expert witness. Lying under oath is in general a bad idea, even in order to save two well-meaning but hapless crooks. There are always alternatives to lying. An expert can decline to testify, a judge can recuse oneself, citing a conflict of interests or beliefs.

If your main point was about whether the means justify the end (consequentialism vs deontology), it has been discussed on this forum ad nauseam, I don't see what you are trying to add to it.

Given the expected long-term value of John's credibility about judging honesty, my guess is that "don't lie about my judgments of honesty" is a good policy for John to adopt.
Given the expected long-term value of Bob's limited ability to manipulate African law without getting arrested, my guess is that keeping Bob out of jail a little while longer isn't worth enough to justify violating that policy.
So given the forced choice you present between (a) lie to his colleagues about Bob and Carol's honesty, or (b) tell them the truth, I'd say John should tell his colleagues the truth.

If judges — as a class — habitually followed their own consequentialist conclusions rather than the law, then statute law would not be very relevant.

Would you apply similar logic to a police officer deciding whether to enforce the judge's decision? What about a general deciding whether to stage a coup against a democratically elected president?

Should John break his oath and tell his collegues that Bob and Carol are telling the truth?

But he's not breaking his oath by telling other judges that court participants are lying. He's not ruling on a case. He's not acting as a judge. He expressing what he has every reason to believe is an accurate opinion.

The other judges may be violating their own oaths by taking what is in effect testimony outside the court room. One could argue that John is breaking his oath by participating in their oath breaking, but I'd say that's at most a second order effect.

I think you've made a complex case a multitude of conflicting values that aren't decided on a spreadsheet, but within a person's head. Different people will weight those conflicting values differently.

With regard to the general question described in the title, there's actually a huge literature. Just for example, Richard Posner's Economic Analysis of Law (and pretty much most of what he's written), and Philip Hamburger's *Law and Judicial Duty."

For what it's worth, in America at least, there is no "state without juries," but there are bench trials, for example when a criminal defendant waives a jury. In that case one -- and only one -- judge acts as both the arbiter of law and the finder of fact.

You only get a panel of judges at the appellate level. This hypothetical suggests that the facts have been addressed by more than one "previous court." But appellate courts essentially never review findings of fact, whether by juries or by judges of bench trials. Neither do they take new evidence.