No, this did not happen. It is a myth. Among the problems with the story:

1. There are no first-hand accounts of this happening.
2. It was never mentioned by black writers of the time who documented lynchings and other horrors done to black people.
3. If you believe it is true, you also have to believe black mothers cared so little about their children that they would let them be used as alligator bait for $2.
4. People did not hunt alligators with live bait. The standard bait was a chicken that had been dead for several days so the smell would attract alligators.

https://willshetterly.medium.com/black-babies-used-as-gator-bait-history-versus-folklore-94234229ae31