The main issue as I see it is: do you think you could obtain and afford better medical care?
If the doctor you’ve seen is realistically the best you’re going to get, I think you should simply follow their advice. I think you need a doctor to make a solid diagnosis because your blood tests are not typical for a gout patient. So I think there is a chance you might have a different problem. If you follow your doctor’s advice and it doesn’t help, the doctor should then be able use that information to move on to other treatments or order tests for conditions other than gout. But if you decide to be your own doctor, what will you do if your new diet doesn’t help you?
If on the other hand you want to see a different doctor before starting a new drug, that makes more sense in my opinion.
One problem with changing your diet is that it could possibly take a long time for the effect to become clear, especially if you are malnourished or losing weight. How will you know if your new diet is working? Can you get frequent blood tests? Are you willing to purchase your own testing kit? And how long are you willing to wait?
Your blood tests for uric acid are low enough that if gout is indeed your problem, you might well be able to manage it without medication. But taking Miluric for a while would make the transition easier for you. Once you are free from gout symptoms, you would be able to stop taking the drug and see if a healthier diet is enough to keep your uric acid low.