I can’t help you with the reflux other than to say that, like gout, it’s sometimes a misdiagnosis. Unless an examination has demonstrated that reflux is indeed the problem, it could be a stomach or gallbladder issue for instance.
Assuming you really are suffering from gout, it sounds like you’ve only had one uric acid test. If so, get it tested again!
You might well be able to win this battle with diet alone in the long run, especially if you count weight loss as part of diet. It would require patience however. Most of the information about gout and diet out there is also quite bad so make sure you aren’t wasting your time by trying to make the wrong changes to your diet.
Meds could be a great help while you are losing weight and ajusting your diet. The important thing, considering your liver issue and the 6.3 uric acid test (I assume you’re not misreading 0.63 mmol/l as 6.3 mg/dl which would completely change the picture), is to stick to very small doses of allopurinol (and if your doctor wants to give you febuxostat which is also known as Uloric, the dose you ought to try first should be no higher than 10mg which might well be a problem depending on where you live). You could also try completely different drugs or indeed foods known to lower uric acid but the nice thing about allopurinol and febuxostat is that they work well enough that a very small dose could well be sufficient. Even with common foods known to lower uric acid, you’d have to consume so much that it might be hard on your liver for all I know.
Possibly you might also get results by lowering your ferritin. See if that might have something to do with the fatigue you’re experiencing. But you should also find a way to lower your uric acid.
So far I have only been talking about drugs acting on uric acid but there are also drugs you could take to alleviate your symptoms. Unfortunately the drugs most people take for gout symptoms often cause liver function problems and you often need large doses for gout. If you can’t use regular anti-inflammatories because of your liver (if you haven’t tested what that does to your liver, I would strongly recommend against testing a regular anti-inflammatory at the same time as a drug which lowers uric acid), one thing you could do is to take a painkiller to help you sleep (keeping in mind you need to avoid anything containing paracetamol/acetaminophen).
If your doctor suggests colchicine instead of an anti-inflammatory, be aware that is also known to trigger liver function issues. They’d have to look into less common gout drugs if they want to suppress inflammation without endangering your liver.