From what you said, it sounds like you need to clarify that your problem is not just fats but almost all foods! Not even broccoli and green beans?! A person who can only tolerate chicken breasts and sweet potatoes has a definite problem, as you are well aware.
You mentioned you had tried many diets, I wonder how many types of cleanses you have done, of what nature, for example the master cleanse? Colemas? Parasites? Have you considered candida? Besides liver and suspected intestinal problems, it sounds like you might have something living in you. You should ask Lapis for advice. In the meantime, continue liver flushing, and take lipid digestive enzymes anytime you eat fat. Hope you can diagnose the exact nature of your problem.