Thanks, those are interesting points.
I'm reasonably sure that a board of naval officers could rule on whether a crime had been committed. It is less clear as to whether they were competent to hand down a sentence of death. But as Hamilton calls it a court-martial, perhaps Nelson used board of naval officers as synonymous with court-martial.
That's what is bugging me at the moment.
Was the court authorised to hand down a death sentence?
Nelson could have sent the prisoner to Procida, as Hamilton suggested, as a fully-authorised judge, appointed by the king, was operating there. He could have sent the prisoner to be tried by Ruffo's judges, authorised on 15 June. These had been busy since February, trying republicans captured by Ruffo's army but were properly established in the capital on the 15th. But, instead, Nelson established a board of naval officers, Neapolitans, to pass judgement on a British ship.
You are right he did things in a hurry, The king only arrived on 10 July, nearly two weeks later. But there should have been no rush, the leading republicans were kept on Nelson's ships in irons and then tried by royal judges in the next year or so. Caracciolo could have been too?