regarding a pirate state in the Caribbean, the most likely period would be during either the ARW or the French revolutionary/Napoleonic period as this is when the major powers are distracted. Once these wars are over the pirate state has a short shelf life however
In terms of historical pirate states there are:
Knights Hospitallier
England
Barbary States
Lordship of the Isles
There are other non-state pirate powers including:
Vikings
Various west coast Irish clans
Pirate of the Caribbean (hey there were some!)
