I believe your second paragraph is incorrect. The purpose of the Atkins diet is to get your body to be in ketosis. If you do not eat a large amount of protein, but instead eat more fat (which is encouraged on Atkins--perhaps more encouraged on more serious low carb diets) your body will not really be getting it's energy from protein. Ideally (though Atkins does allow you more flexibility), you should eat enough protein to maintain your muscle mass (or to build muscle if you are doing resistance training), and no more. The rest of your energy is supposed to come from the fat you eat and your body fat (plus a little from the few carbs you do eat). This idea is more emphasized in some other low carb diets.
BTW, I was not claiming that I was on the Atkins diet while on a fast. I was claiming that if I had an Atkins or stricter day on the non-fasting day, it would not interrupt ketosis. Since Atkins gets you into ketosis and keeps you there if strictly followed, someone who was already in ketosis should remain in ketosis when shifting to an Atkins-like diet.
I fear you are correct about the restarting digestion and delaying healing.
Never the less, I have repeatedly heard rabbis forbid fasting on the Sabbath. The Sabbath is the most important holiday of the year, for all that it happens 52 times a year. It is supposed the be a day of rejoicing, and not only are you not supposed to fast on the Sabbath, you are not supposed to intreat or supplicate God on the Sabbath -- ordinarily good things. Instead you are to give God thanksgiving and praise. I have heard this many times, but it has now begun to bother me that I have not been following this doctrine. I want to be sensitive to God's Spirit, and I have begun to think that perhaps He would not wish me to fast on the Sabbath, though because of your reasoning, I have done so in the past.
My purpose in posting my question was to find a way to honor God while minimizing the disruption to the healing process. Perhaps it is futile, and I will just have to resign myself to the delayed healing. I hope that someone will be able to help me in some way other that to ask me to not follow my conscience.