I really like the firing mechanic you made, and I think there should be an option for a health based battleship match and a time based one, like the one you already have. I feel like the time based one can drag on sometimes, and health based ships would make things a little more exciting.
As for the shop/upgrades, I think it's fine as is. Players should be able to upgrade their ship AFTER the end of the match.
And I agree with Binkus on adding challenges; you could have multiple small enemy ships moving side to side and try to hit them all. Score can be based on who can destroy the ships the fastest.
Overall, good job on the game!