I don’t know anything about Down Syndrome and gout.
Obviously you want to be especially careful with potentially dangerous drugs such as allopurinol, especially if your son is not able to give reliable feedback on any side effects he might experience. But I don’t know anything specific which would make allopurinol unadvisable. Frankly, I would be more concerned about what colchicine could do to someone with Down syndrome if they were to take it for too long.
But I can tell you this: you can’t judge whether the steps you took to prevent gout were working or not based on whether or not attacks occurred the next day or even the next month. Indeed, effective treatment often triggers attacks during the initial phase. In some cases it can take years of treatment for the attacks to stop. Since your son has only had gout for a few years, there is reason to hope effective treatment might cure his gout in 6 months or even less. But in a single month? That’s not likely.
Besides rare and expensive modern imaging technology, the best way we have to guide treatment over a reasonable timeframe (weeks to months) is blood tests. The value you want to look at is the amount of uric acid in the blood. I would recommend you look at your own test results as well as your son’s because, if you have gout, it can develop without causing flares for a while and you might end up being sorry you didn’t take more aggressive steps to prevent your gout from festering. If you posted test results here, we could help you interpret them.
Also, the dietary information you used may be wrong. There is a lot of misinformation out there. Not that it matters much: drugs are easier and work much better.