Nobody was really interested in whose fault this thing was. Klingons probably don't even believe in "war crimes" or "innocent bystanders", and from the get-go they publicly admitted that they had no case and no interest in having one; they just wanted to exercise their right to pester Worf.
It was a win-win scenario: if the UFP pointed out that the Klingons were in the wrong and should just shut up, this would be a propaganda victory for the Empire, and if the UFP extradited Worf, this would be a propaganda victory for the Empire. The only reason the Klingons didn't gloat more at the end was because their duplicity in organizing the incident was a tad too embarrassing to be made public - but make no mistake, they won this round. In their own minds at least.
As for why they even competed, the stated reason must be bogus. The trial did not shame the UFP into ceasing aid shipments: as the trial started, the shipments were already ending all on their own. Worf had been entrapped on the second-to-last sortie, after all. So a more general propaganda victory must have been the aim, and it depends on the Klingons and their intended targets (Romulans? Cardassians? Minor UFP arch-enemies, or cultures currently undecided on whether to become such?) whether the victory really was achieved. The UFP and the outcome of the trial as such would have no effect on that.
Technically, the Klingons might well use cloaks on transport vessels. After all, that would give them strategic advantage: if the enemy didn't know where the logistics were going, it couldn't know where an invasion would be coming from, or even whether an invasion were coming. But that would mean cloaking all logistics at all times, especially in peacetime and in directions where Klingons had no military intentions.
Timo Saloniemi