I think something like an adversarial system with a jury of peers would still be a good choice with perfect Bayesian agents. This structure exists largely to avoid conflicts of interest, not merely to compensate for human irrationality. Unless you are assuming that perfect Bayesian agents would not have conflicts of interest (and I don't see any reason to suppose that) then you would still want to maintain those aspects of the legal system that are designed to avoid such conflicts.

There are two common classes of reasons why evidence may be inadmissible or excludable in a trial. One of these classes should probably be admissible with a perfect Bayesian jury, the other not.

Evidence that is inadmissible because it would prejudice or mislead the jury (like information about prior convictions) would probably be fine with a jury of perfect Bayesians but evidence that is thrown out because it was obtained in some way that society deems unacceptable might still be rejected because of broader concerns about creating inappropriate incentives for law enforcement.

This raises the question of just how perfect your Bayesians are however. If they are very good at correctly weighing relevant evidence but still have computational limits these concerns would probably apply. If they are some kind of idealized agents with infinite computational capacity then you might draw different conclusions but as this case is impossible it is not very interesting in my opinion.