A strict low purine diet can lower the blood ua level by about 1 mg/dl. It's unlikely your new diet alone actually lowered the ua level from 8.1 to 6.3. Have ua tests periodically (every 6 months?) when there is no gout attack. They can assess the ua level more accurately.
Two factors can increase the chance of having ua kidney stones --excreting more than 600~700 mg of ua in urine a day and the urine ph is less than 5.5. You can test the ph of the first morning urine yourself. But it may need a doc's prescription for a 24-hour urine ua excretion test. It's unlikely the temporally increase of the ua excretion during a gout attack can actually cause kidney stones. You can increase the urine ph by avoiding acidic food and drinks, taking mineral supplement, and drinking baking soda in water.