My response has nothing to do with batteries or kayak, but being a canoer in Indiana, I've often thought about putting a trolling motor on my canoe. I have outriggers and with those outriggers, the canoe is as stable as any fishing boat, even can stand in it.
But I've never done this for one reason. Indiana has a law that say if you put a motor (even a trolling motor) on a boat, it is now a "motorized vehicle" and it has to be registered in Indiana same as any other motorized boat.... even a trolling motor.
Wind powered or people powered boats do not have to be registered, but if you have a sail boat with a motor for back-up power, it has to be registered and have a tag number.
I don't know what the rules are in your state, but please check into it first.