I was hoping the battleship would win---battleships were the best AA platforms of WWII, which would help us a bit, I imagine, given that our superior tactics can lure enemies over the battleships below. They were also, by war's end, pretty much unsinkable by destroyers and MTBs assuming any degree of competence (or radar. Radar covers a lot of mistakes for warship tactics) on the part of the battleship's captain and his escorts. The battleship gives us a solid advantage using tech that was all finalized in 1939 (year the Iowa-class was laid down), in fact using a worse version of the techs involved in the Iowa-class. Almost all of which we already have. The battleship shouldn't be difficult and we should get everything we ask for, simply because apparently we can play in the 60s (well, the Cannalans can, anyway) and this is a much simpler (if larger) design. The missile would give us a tremendous advantage IF we get the 50s/60s tech down in the design, by no means guaranteed. However, if the missile is going to win, let's do it right.
Also, Kot's missile seems a bit feature-heavy. We don't really need to replace the engine, stripping the thing down should reduce the weight enough to give us the speed we need, HEAT is a terrible way to kill warships unless you always hit the turrets/somewhere else that's full of high explosives, and it shouldn't have different variants. ASM ONLY, don't try to make an ASM and a cruise missile all at once. Also, you should mention that there's a delay in the tracking system (can be set before launch, obviously), so it doesn't try to track friendly vessels if launched on a vessel in the middle of a formation.
EDIT:
A few notes,
Don't use a multi-stage design it's adding complexity for very little gain. Just mount the rocket boosters on the side of the missile. So when you hit the fire button all it has to do is ignite the boosters and have a kerosene starter ignite the turbojet. Versus igniting the rockets, waiting until it's empty, using something to decouple it from the missile and, then start the turbojet.
Do use multiple stages. Separating stages is as simple as popping a black powder charge roughly the size of your thumb. Heck, you could probably manage to separate stages just by using gas leakage from the first stage (assuming you're using a solid rocket motor, all you'd need to do is drill a tiny hole or set of holes around the edge of the rocket motor's casing. When the burn finishes the fuel, the holes will be uncovered and the flames/heat can ignite a black powder charge to super-pressurize that compartment, pushing the first stage off (and igniting the second stage if necessary).