I heard about the problems people have been having getting the black ship early in the game.
Still on my first campaign on normal with Shimazu, but this is what I've found:
I sent 2 medium bunes and 3 bow kobayas against it and auto-resolved. Lost badly and two of my tattered ships limped away. Then, since the ship was near my trade node, I mobbed it with 10 trade ships. The "advantage slider" thingy that shows the balance of forces was nearly all yellow, 90% to me. Auto-resolved and only lost a few archers with almost no damage. It seems the Black Ship can be won easily by attrition.
Just to test this, I reloaded and added 5 trade ships to the initial fleet I sent, so a 10 ship stack with 5 trade, 2 medium bunes, and 3 bow kobayas. They got destroyed. Sent in the remaining 5 trade ships, they got destroyed too.
It seems a handful of cheap battle ships and then a mob of trade ships over 2 attacks in the same turn works really well. Got the black ship and only lost 3 very cheap vessels.
edit: oops maybe this should be moved to "Strategy"