Ketosis is about the carbs, not the fat. The Atkins diet is way higher fat than IP and they go into ketosis, too. There is even a ketogenic diet for seizures that is mostly fat.
Don't be overly afraid of fat. We need SOME. Be sure to put your teaspoons ON your food!
Losing weight is hard. Maintenance is hard. Being fat is hard. Pick your hard. (-Ishbel)