Okay, so. My last plan didn't work so well.
Why?
Basically because I'm a terrible gardener. I'm pretty confident that if I'd watered the grass everyday, both to water the grass and to dilute the urine, this would have worked great. Neglected as it was this lasted for like 2 months (watered every few days...). Eventually all the grass died, mostly because I slowly stopped watering it. Poo was hard to pick up out of the grass because it got pretty long, so I think a flatter groundcover would have been smarter (Stepables, anyone?).