Many of your issues could be due to gluten. Perhaps not all, but celiac issues can and do skew liver enzymes as well as cause a host of other 'unusual' problems. Try clearing up the celiac first and then go from there.
Why do you disagree with his diagnosis? Actually, I think he may be on to something & you are lucky to have found a dr. that is aware of Celiac -- a diagnosis of Celiac takes an average of 11 years because it is so tricky to diagnose.
Celiac has such a weird assortment of symptoms & can manifest itself differently in different people. It can effect so many different systems that it can either look-like or lead to other problems like thyroid issues, or diabetes, or even neurological issues. Some of your symptoms sound a lot like what I had, and it is confusing because they all seem unrelated to one another. I had weight gain (15 lbs., which fell right off when I went gluten-free), skin issues like red/rashy skin, depression and anxiety, and insomnia & lots more. After I was gluten-free for 11 months, my health really turned a corner & now I feel fabulous.
Anyway, give it a try & work with you doctor. He may have found the source of your problems. If it is Celiac & you don't go gluten-free, it could lead to some serious health problems later, like increased risk of cancer.
Good luck & I hope the gf diet works for you.