The water level will fall.
Work Step by Step
The iron in the battleship will displace a greater amount of water while floating than when submerged. A floating battleship of weight w displaces a weight w of water, which takes up more space than iron of weight w. When the iron in the ship is submerged, and all the air pockets are filled, it displaces only its own volume, so the water level drops. Try it yourself in your kitchen sink, with a deep heavy bowl playing the role of the floating ship. Overturn it to "sink" it.