I seem to recall enemies of the crown being a higher priority than pirates, but I don't have the book in front of me to confirm. I'll check it tomorrow once I return from Wonder Con.
No, that is last. First priority is pirates with a bounty from the Naval ships own nation.
Then as OP writes, the second is if bounty from other nation.
I see the thematic reason for not leaving you alone during war, is that you are just a criminal. Without a letter of marque, you might start attacking any nation as far as they know.