It sounds like you've done some homework. The general idea of the organics for this is to have a sufficient population of microbes on hand for any dog pee event. The table sugar should be about a handful applied on and around the edges of the browning grass. The sugar puts the microbes into overpopulation mode so there will quickly be enough to decompose the urine. If that is what you've done and the spots didn't go away, you might look into a soil test (Logan Labs, $25). There might be a chemistry reason why the microbes could not handle the extra urine.

Or it could be the zoysia is less tolerant of the additional urine than other grasses and immediately goes into dormancy. I find zoysia to be a problematic grass to grow. It is higher maintenance than they want you to believe, and very unforgiving if you make a mistake.