Avocados, organic cold-pressed olive oil, coconut oil, or flaxseed oil should be part of your daily regimen to avoid losing too much weight, but it's inevitable on any diet that removes wheat, Sugar and alcohol. It's your call to make about animal fats, but fish that does not have mercury is going to be your best bet. Anything baked in oil is definitely preferable to fried. Without fats, your brain will not function properly, and I've heard anecdotes about it resulting in severe depression.
Also, you could try fattening up before trying out the diet as well if you're really concerned. Just don't make your condition worse in the process. It's a fine line to toe.