I heard Eddie from The Courtship of Eddies Father chanting this while on a hike. Googleing around I found a water rat is another name for a muskrat and Eddie was saying
“Fat fat the water rat, fifty bullets in your hat.” So I’m assuming you kill a muskrat and make a hat out him?
See this looks to be referenced in Family Afffair but I’d bet origin is quite a bit older as it seems to be a popular schoolyard rhyme prior to the show using it.
Interesting how The Courtship of Eddie’s Father (where I heard it) and Family Affair (where another poster heard of it) were both late 60s / early 70s shows.
I wonder if they had the same writers or anything. Odd both shows would feature that chant. I don’t think The Courtship of Eddie’s Father was set in NYC like Family Affair so it can’t be a local rhyme.