Maybe you don't need a battery and can go with wireless power transmission. A scaled-up version of this : https://www.youtube.com/watch?v=UQ3K0suY1Dc https://w

Maybe you don't need a battery and can go with wireless power transmission. A scaled-up version of this : https://www.youtube.com/watch?v=UQ3K0suY1Dc https://www.youtube.com/watch?v=UQ3K0suY1Dc To provide more power to the led you probably need a bigger coil inside the eye, it should work if you hide the induction transmission loop inside a hat.