The flushdeck frigate can certainly survive escaping from a fort. You've already done it. It's the same type of ship as the one in which you had to escape from Speightstown with the fort firing at you.
And it's certainly capable of capturing the enemy frigate and then escaping to the lighthouse because I've done it, numerous times. You're in trouble if a lucky shot takes down your mainmast, so it's wise to save game just before putting to sea. (In fact, it's wise to do that routinely anyway. It gives you a place from which to try again if something goes wrong with a quest, the game crashes randomly, or anything else unpleasant happens.)
And it's certainly capable of capturing the enemy frigate and then escaping to the lighthouse because I've done it, numerous times. You're in trouble if a lucky shot takes down your mainmast, so it's wise to save game just before putting to sea. (In fact, it's wise to do that routinely anyway. It gives you a place from which to try again if something goes wrong with a quest, the game crashes randomly, or anything else unpleasant happens.)