The Atkins' Fat Fast is similar to some of the diets for epilepsy.

Here is a link to The Charlie Foundation, which has information on variations of the ketogenic diet for epilepsy. Perhaps something there will answer part of your questions.

https://charliefoundation.org/diet-plans/

I have had long periods of time, months at a stretch, of being on very low carb, very high fat, adequate protein, and was able to greatly reduce migraines with that regime. Sometimes with greater amounts of food overall, sometimes with not so much. Getting the ratios of PFC seems vital for keeping the migraines away, for me.