I am no expert either, but a couple of thoughts:
Coconut oil is mostly MCTs (like lauric acid), that dont require bile, so that may be okay. MCT oil would be much better, as you need some fats, and some liver diet websites recommend it.
Agree on the fructose, that is hard on the liver, so you may want to limit fruit, and if you want sweet use glucose as a sweetner (no liver processing, unlike fructose containing sugar and honey)? Up to you...
Vege oils are also hard on the liver, at least according to some websites, but not sure whether thats accurate or not?
Dan shen, and other hepaprotective agents might be worth looking into. Herbs like this actually prevent liver damage, and dan shen (chinese herb), is the most potent one I know of. Id say this herb is FAR better than liverwort/milk thistle, which I read can actually put extra load on the liver. Give it a google if your curious.
All foods are processed in the liver.
My quick googling made me question your doctors advice a little. He may still be right, but:
Vitamin K for example, is a vitamin vital to liver function (found in cranberry juice in resonable levels, along with choline, another liver health promoter), is a fat soluble vitamin. Clearly, youll still need some fat, even if you are limiting it, otherwise youll stop getting fat soluble vitamins.
This website recommends a moderate fat, high carb, low protein diet, low salt diet for liver disease:
This website also recommends, medium fat, high carb, low protein:
This website also recommends medium fat:
Its appears, that if your liver is not having specifically bile problems, you want to limit protein more than fat (I am not a doctor, its just what I am reading. If you have bile or bile duct problems, yes you want to limit fat), and go moderate fat (needed for fat soluble vitamins, including ones needed for the liver, and also "Moderate intake of fat, as prescribed by the health care provider. The increased carbohydrates and fat help preserve the protein in the body and prevent protein breakdown" - from first article above)
If I was you, id research it a bit more and make up my own mind.
I am not sure about your specific condition, or the state of your liver, but what little googling, in a very cursory manner mind, makes it sound like your want high carb, low protein, medium fat, particularly if you can get some MCTs in there for the fat portion.
Whatever you decide macro wise, remember paleo aint about macros, its about real food, foods we are evolved to digest. If it calls for less fat, there are less fatty foods. If it calls for more carbs, their are more carby foods.
Just because someone says "you need to limit fat" or your need to "eat more carbs", doesnt mean you need to start eating processed foods.
Limiting physical excercise is good advice for sure though... (I had hepatatis once and this is a definate. Dont push yourself physically. Also, if your bilrubin is up, sunlight helps get rid of it, which can make u less irritable...which if your bilrubin is up you may appreciate cause the irritability is annoying)
Again, not an expert, make up your own mind, hope you get better!