I wouldn't go that far. There are many cases where the legal system explicitly deviates from Bayesianism. Some examples:
Despite the fact that Demographic Group X is more/less likely to have committed crime Y, neither side can introduce this as evidence, e.g. "Since my client is a woman, you should reduce the odds you assign to her having committed a murder by a factor of 4." (Obviously, the jury will notice the race/gender of the defendant, but you can't argue that this is informative about the odds of guilt.)
Prohibition on many types of prejudicial evidence that is informative about the probability of guilt (like whether the defendant is a felon). (This can be justified on grounds of cognitive bias maybe, but not Bayesian grounds.)
In the US, the Constitutional prohibition on using the defendant's silence as evidence, despite its informativeness, e.g., "If he's really innocent, why doesn't he just tell his side of the story? What's the big deal? Why did he wait hours before even saying what happened? Did he need to get his story straight first?" (Again, the jury will notice that the defendant didn't take the stand, but you can't draw their attention to this as the prosecution.)
The exclusionary rule. The impact of illegally-collected physical evidence (i.e. not forced confessions but e.g. warrantless searches) has a small to non-existent impact on the evidence's strength. The policy on excluding illegally-obtained evidence may be justified on decision-theoretic grounds, but not on Bayesian grounds.
Outside of trials, the fact that you have to wait years before you hear a judge's binding opinion on whether or not a law actually can be enforced (i.e. is Constitutional).
You give the legal system way too much credit.
The policy on excluding illegally-obtained evidence may be justified on decision-theoretic grounds, but not on Bayesian grounds.
In that case, why should we design the system on Bayesian grounds?
I think that's really why I concur with komponisto - our system may not be optimal, but optimal for a system has to work as a system, including resistance to gaming. Aside from what you suggest about constitutionality, on which I have no comment, your changes are generally unlikely to improve the ability of a legal system to prosecute the guilty and acquit the innocent.
In 2004, The United States government executed Cameron Todd Willingham via lethal injection for the crime of murdering his young children by setting fire to his house.
In 2009, David Grann wrote an extended examination of the evidence in the Willingham case for The New Yorker, which has called into question Willingham's guilt. One of the prosecutors in the Willingham case, John Jackson, wrote a response summarizing the evidence from his current perspective. I am not summarizing the evidence here so as to not give the impression of selectively choosing the evidence.
A prior probability estimate for Willingham's guilt (certainly not a close to optimal prior probability) is the probability that a fire resulting in the fatalities of children was intentionally set. The US Fire Administration puts this probability at 13%. The prior probability could be made more accurate by breaking down that 13% of intentionally set fires into different demographic sets, or looking at correlations with other things such as life insurance data.
My question for Less Wrong: Just how innocent is Cameron Todd Willingham? Intuitively, it seems to me that the evidence for Willingham's innocence is of higher magnitude than the evidence for Amanda Knox's innocence. But the prior probability of Willingham being guilty given his children died in a fire in his home is higher than the probability that Amanda Knox committed murder given that a murder occurred in Knox's house.
Challenge question: What does an idealized form of Bayesian Justice look like? I suspect as a start that it would result in a smaller percentage of defendants being found guilty at trial. This article has some examples of the failures to apply Bayesian statistics in existing justice systems.