Dr. McDougall says that if you're having a hard time losing weight following the regular Starch Solution, which allows for a limited amount of healthy plant-based fats, that you should avoid them. You've already eliminated animal fats and oil, so this refers to healthy fats such as nuts and nut butters, seeds and seed butters like tahini, avocado, coconut, and soy products (they're surprisingly high in fat!).
These foods have a high calorie density, so they're easy to overeat. A two-tablespoon serving of peanut butter is 188 calories. You could eat six cups of broccoli for the same amount of calories, which would fill your stomach more.