Cool idea! To reduce traffic on the blockchain, I think you only need to create one or two transactions for the hash of both players' starting position. The players can submit their turns through a website or a P2P communication channel.
Detecting cheaters afterwards would be a reasonable compromise, since you'll probably want to play Battleship more than once on the same website, or more than once with the same person.
In fact, I'm not sure whether a DPOS blockchain would be needed. OpenTimestamps might be efficient enough, while taking advantage of Bitcoin's security.
Nice! These are great ideas, but you've somewhat jumped the gun. There is also value in not lowering blockchain traffic. There isn't a lot users can do here at the moment. Users need something to spend RCs on. It's more useful to force players to pay to play using resource credits rather than host a dedicated server.
Also, if you are trusting a third party server this isn't secure. The server can lie and there is no way of knowing if a player made a certain move or not. At this point you'd have to use second layer scaling solutions where players were signing transactions with their private keys off chain. This extra layer of complication isn't something that should be worked on unless the game actually went viral.