I'd say that keto is not only much closer to Atkins than what you're doing--but that it is Atkins. At least, it's within the range of protein/fat intake that Dr. Atkins advocated. What you've been doing is more like Stillman's, or a protein sparing modified fast.

Adding fat might slow weight loss, it's hard to say. But does weight loss really need to be this fast? Do you really need to lose almost a pound a day? I'm not judging here, sometimes a water fast or modified fast can be therapeutic, but it sort of depends.