No medical studies that I know of, and at 1,000 calories a day, I doubt the Atkins Fat Fast would have any of those benefits. If a water fast is too daunting, check the Time Restricted feeding studies, two good interviews (first one was in 2016) with Satchin Panda on Rhonda Patrick's website. (Spoiler, even black coffee breaks a fast)
https://www.foundmyfitness.com/episodes/satchin-round-2