So who would win a battle between an electric eel and an alligator?

Well, actually a smart alligator would leave an electric eel all alone.
Nature provides all sorts of defenses – poison, camouflage, big brains combined with opposable thumbs. For the eel, it’s a shock of electricity that can save its life from a predator.
Here’s visual evidence of an entanglement between the two captured by a Brazilian fisherman.
The eel’s effort was more successful than this python which tried to eat an alligator.
Photo credit for arm-raising alligator – Funny Junk