Do you think it's the sweating or the yoga exercises that are helping you keep gout in check? I would think it's the sweating.
Various toxins, uric acid among them, are excreted not just through urine but also through perspiration. Since we generally don't sweat that much, I would think the urine path is usually much more effective. However, if you engage in a sweat-inducing activity or spend time in a hot/humid environment, then you can get rid of a significant amount of liquid (and presumably a significant amount of UA).
Instead of Bikram yoga I've been going 2 or 3 times a week to a gym where they have a dry sauna, a steam room and a whirlpool spa. I have weighed myself: a half hour of sweating "like a pig" in these facilities drops my weight (currently I'm about 170) about 1.0 or 1.5 pounds. It's all sweat and when I drink water afterwards I regain the weight loss (don't want to be dehydrated -- can bring on a gout attack!). I still don't know how much uric acid leaves through the sweat but *maybe* sweating is helpful.
Also, I have done other things to try to prevent a gout attack (lose weight, change diet, use supplements, etc.). So far, so good. I have not had an attack for a couple of months now (way too short of a time period to make any valid conclusions, I know).
Still, the idea of uric acid excreting through perspiration makes a lot of sense. Unfortunately I have seen very little information on the Web about this.