Jesus of Nazareth was a Jew, but he fasted for 40 consecutive days in the Wilderness: so did Moses and Elijah.
The question I would ask myself with regard to your question is: is it Biblical or taught by the Torah and/or the Tanakh to fast on the Sabbath, or is this something concocted from men and Rabbis generally?
The Atkins diet does bring the body into ketosis yes, as will any low carb' diet, but this is only "partial-ketosis" and not "full-ketosis".
Only water-fasting brings the body into full ketosis when full fat metabolism is reached.
Any feeding during a fast interrupts the digestive energy from its main task of healing and cleansing.