When you say you get mucus from fat, what fat are you referring to?
Mucus is also a symptom of serious dysbiosis, and what you really need is to restore your bile flow, and to restore your intestinal microbiota.
While the Liver Flush may not be a pleasant experience the first few times you do it, on the long run, after a few months, Liver Flushing will improve your digestion and your health, to the point where most of your food intolerance will be past.
If I were you, (and trust me, about 20 years ago I was just like you, extremely intolerant of all dairy, eggs, meat, flour, sugar, .... many fatty foods ) I would research the Fecal Transplant therapy and I would try to find a donor in my immediate family.
I believe that you already know a lot about diet, so as the time goes, and as you improve your liver health and your microbiota, you will have to change your diet, and include many tings you have problems tolerating now.
I personally do not believe in industrially processed supplements, and are more oriented toward natural sources of vitamins and minerals.
Natural super-foods are seaweed (all minerals are there) , unrefined Sea Salt (all minerals are there), vegetable juices (almost all vitamins are there), animal liver (huge super-food rich with almost all vitamins and minerals)
Some of those foods you may be intolerant of, so, it will be your future diet once you fix your liver.
Do not be too hanged up on old books. Many were written before the invention of Antibiotics , before the mass vaccinations with 10, 20,40 vaccines .... just because old health teachers were unaware of liver flush, or FMT , it doesn't mean you should not study it yourself and use it in your own advantage.
Good luck, and do not be dogmatic about anything.
Dogmas are not good for health. Health is all about change. Stay too long on any restrictive diet, and you will achieve the opposite of what your goal is. Your goal is health. Extremely restrictive diets, on the long run, lead to poor health.