The problem with this approach is that it's a quick, temporary "fix" that doesn't work long-term and can actually cause harm. Permanent, healthy weight loss requires permanent lifestyle (and mindset) changes. Especially for someone who is as overweight as Kate, where it's not just a few vanity pounds.
Eating a lot of animal products is unhealthy regardless how much water you drink. The book How Not To Die, written by a physician, cites over 2000 studies on nutrition and diseases (there are many other books on the subject, but this one seems to be the most rigorous in terms of research). I don't know of any studies specifically that compare "extremely low carbs diets versus being fat", because that seems like an arbitrary and unscientific constraint, an apples to oranges comparison if you will - there are many ways to not be fat that don't involve low-carb diets.
A diet that includes fruit and vegetables would not be low carb, as fruit and vegetables are mostly made up of carbs.
I think the most important thing is to separate hype from science. None of the fads like Atkins, Keto, etc., are based on long-term, rigorous research. They lead to quick short-term weight loss in some people, but the same can be said about chemotherapy. Just because something causes weight loss doesn't mean it's healthy.