The immune system is not only what causes the swelling and pain, it also plays a role in the formation of crystals in people suffering from gout. Though that is not the only factor, the more you have gout symptoms, the easier it will be for crystals to form.
In order to take this further, I need to rely on speculation rather than evidence. But it seems reasonable to me that, after a break from gout symptoms, the immune system might both contribute less to the formation of crystals and be less likely to react to any crystals which might form anyway. This would explain an otherwise odd fact: some people who have cured their gout properly can let their uric acid climb a bit without getting gout symptoms. Sometimes it takes years with unsafe uric acid levels for gout to come back.
On the other hand, looking at how long vaccinations can remain effective, I suspect that your immune system isn’t likely to go all the way back to its pre-gout state. And indeed, while most people with unsafe uric acid levels don’t have gout, gout-prone folks get symptoms over and over again with the same uric acid levels.
I for instance have had unsafe uric acid levels for more than 10 years before developing gout-like symptoms (I don’t know for sure which of my early symptoms were actually caused by gout, if any). Now I take care to keep my uric acid much lower than it used to be (since you ask, I use an alternative to allopurinol and my overall experience has been a bit of an ordeal thanks to terrible doctors).
Your doctor is right that UA levels take a while to normalize.
It is also a fact that they fluctuate. So no matter how long you wait, you will not get very useful data unless you test your UA several times and average. At the same time, the higher effectiveness of allopurinol (as compared to diet for instance) means you can get a pretty solid hint of how the therapy is going to play out with an early test. This is only part of why I would recommend testing early. The main reason is simply that you need a blood test for your liver and kidney function anyway so you might as well get your UA tested while you’re at it (the extra test would take very little effort as compared with drawing your blood and so forth).
I would recommend you start tweaking your diet before going off allopurinol (but you could perhaps take a lower dose at that point). You can get an idea of how well your new diet is working even while you are still on allopurinol. If you were to take allopurinol for much longer, it wouldn’t be a big deal but three months is really short so you better have an effective way to lower your UA ready the day you quit if you stick to your 3-month plan (something I can’t recommend honestly) and ajusting one’s diet takes time.
In any case you want to take a fairly strong allopurinol dose unless you are pretty confident the attacks are gone for good. So this talk of discontinuing allopurinol is premature. First see if you get any more attacks or if that was the last one.