Battleships and a dedicated CAP at Midway

Would battleships and a stronger fighter patrol have saved the Japanese carriers at Midway?

They could accelerate Japan's counterstrike and inflict more damage, but probably would not prevent the carrier losses.

The proposal places Japanese battleships with the carrier force and keeps Zero fighters on combat air patrol instead of concentrating on rearming strike aircraft. That arrangement could make the fleet react faster when American aircraft appeared. It might also let Japan launch its own carrier strike earlier and damage the American carriers more seriously. The Japanese carriers would probably still be caught by surprise and lost. The change raises the cost of the American victory rather than guaranteeing a different winner.