You're a naval officer. Your job is to hunt pirates!
If you want the corsair refit, you will need to pay the governor of La Tortue to make peace with them for you. And you'll need to shirk your job as a pirate-hunting naval officer until you've been to Nevis to get the refit.
Nevis is indeed the only pirate port. In earlier periods, Turks Island is pirate, but in "Revolutions" and "Napoleonic" it's British. There are other pirate colonies but they don't have shipyards.
Now, if you were a privateer, you would have another choice. You could raid Nevis, demand money from the governor, loot the store, and get your ship refitted while the shipyard owner is scared. But you're a naval officer and a loyal servant of the USA. So if you raid Nevis, you're not allowed to loot the place, you have to capture it, and then only for the USA, not for yourself. And of course, if you do that, there won't be anywhere to get the corsair refit...