I'm not an LSO, and I only have about 150 traps, but I have been on a night straight-in approach (Case III) where I broke out at 1/4 of a mile around 180' feet. If you add the height of the carrier, 40 feet, then it was about 140' feet above the flight deck. Probably somewhere from the IM (in the middle) to IC (in close) position. 200 feet is your typical HAT (height above touchdown) for any precision approach (PAR, ILS) or the "lowest you can go without the runway environment in sight." If you don't, then you have to execute a missed approach.
During carrier ops we use what is called ACLS, kinda something like ILS. The mins for a ACLS approach are 200' and 1/4nm but it is really up to the LSOs to how low they want to bring you in. At 3/4 of a mile, if you don't see the ship, instead of calling the ball, you call "Clara Ship." This tells him you don't see the ship or fresnal lens at all. He will more than likely call "Paddles contact, continue." When you see the ball you just call "ball" and the LSO will echo "Roger Ball"
One other thing is that during good wx carrier ops, we fly without our landing light on, but when the weather gets bad (3/4 vis), the LSOs make us turn them on so they can see us even though we can't see the ship.