You can lose weight if your liver is weak due to the fact that bile is necessary for the absorption of fats. If there is little or no bile released, then it is likely that fats will not be absorbed and this can contribute to weight loss. As far as I know the only fat that can be absorbed without the presence of bile is coconut oil so this is a good fat in your situation.
According to a radio interview that I listened to with Andreas Moritz , hydrochloric acid production in the stomach will also slow down when there insufficient bile. Low hydrochloric acid will hinder the absorption of protein. So, not absorbing proteins or fats well could explain your symptoms.
From personal experience, one of my first symptoms of liver congestion is weight loss and after a few Liver Flushes I am finding that I am able to put on some weight even thought I am not eating any differently then before the flushes...By the way I'm also hypothyroid, apparently with hypo everything slows down including liver and gallbladder function, so that may be part of the reason for the recurring liver congestion