I took a look at a web site for artificial turf and they recommend a solution of Simple Green or Nature's Miracle to deodorize their product. They also mention just doing a good hose down too.
I mentioned water and I see that she hoses it daily, but I think I would try more water less often (a thorough flushing once or twice a week rather than a hose on it every day).
If it has a chance to dry out between waterings, the sun and stuff will help break it down.
Vinegar will do the trick! Its a fantastic bleach alternative and not a pollutant either. Dilute with water - 1/2 and 1/2
Since you can use bleach to disinfect a fish tank and to disinfect beer bottles before bottling, so long as you rinse thoroughly, it should be safe.
A really thorough hosing would probably do the trick, though. Dog pee won't kill grass or plants so long as you water it thoroughly within a few hours.