Fat is fine on the day of the flush providing it is combined correctly with other foods. However, the inclusion of fat it "heavy" complex foods can take many hours to digest.
The presence of oleic acid (found in most oils) in the Ileum (where the small and large intestines join) will reduce the biliary reflex.
Consequently if you eat oil at mid day then preferably mix it with protein, vegetables, salad and fruit. It is possible to mix oil with carbohydrate and vegetables but this takes longer. But it is not wise to mix carbohydrate and protein together as this meal takes a very long time to digest and can sit in the stomach for ages. It also depends upon the condition of you bowels. If in doubt follow the guidelines.