I don't think there's any established story with an alligator Loki, I think that's new.

There was frog Thor in the comics after Loki turned him into one:



That was mentioned in the films (Thor: Ragnarok), and the story in the comics did involve alligators (frog Thor lures them out of the sewers with a magic pipe to help the frogs in their battle against the rats), but none of the alligators were actually Loki, they were just alligators.

So as far as I know, I think alligator Loki is just a new variant, showing us that 'variant' covers a lot of ground. But hopefully we'll hear what alligator Loki has to say for themself next episode.