There is the matter of cost which might justify your doctor’s approach but, if that were financially realistic, I would recommend getting the second blood work on the drug sooner.
First, you could otherwise potentially be exposed to quiet but dangerous dose-dependent side effects (impaired liver function for instance) for 3-4 months.
More importantly, the dose ought to be initially set based on your blood tests whereas your doctor seems convinced she is able to divine how much you need. There’s nothing magical about 300mg. If you need more for instance then taking an inadequate dose for several months may cause you weeks of pointless gouty misery. If you should indeed have been taking this drug for 15 years, your body is probably riddled with deep uric acid crystals so it would be prudent to aim for uric acid test results well under the maintenance target.
So, side effects…
There is such a thing as psychological side effects so obsessing about lists of side effects may be counter-productive. Most people do fine on that drug.
Unfortunately, the side effect you are most likely going to experience is gout, gout and more gout. So make sure you have pills to suppress gout symptoms on hand. It seems the longer you’ve been delaying therapy, the longer the gouty cleanup phase lasts.
About the deadly reactions, my understanding is that the risk is greatly reduced if you have no East Asian ancestry. If you do, I understand there is a genetic test you could take before trying the drug.
Anyone can get pretty bad side-effects but they are normally annoying rather than deadly (or least not immediately deadly). I’ve been told the one thing to watch for carefully is a skin rash. If you get a proper rash and not mere itching, stop taking the drug and report to a doctor pronto. Signs of an allergic reaction such as lip swelling are of course also grounds to stop taking the drug and get help. If you get other weird symptoms, as long as they are not serious I’ve been told I could keep taking the drug and see if they go away. Sometimes one’s body simply needs time to get used to changes.
If you want to play it safe, you could start with less than 100mg. That seems to be a pretty safe dose and I’ve never seen a professional recommendation for taking less than 50mg as a first dose but that doesn’t mean smaller doses are actually harmful. In the unlikely event you do react badly to the drug, I think it is reasonable to expect very small doses to be less uncomfortable.