Well, all I can tell you is what I would try:
I'd use 100% natural fertilize and apply it more often, like once a month. If you have someone fertilizing your lawn, I'd think about switching to Natural Way or the like. The natural fertilizers will help promote good microbe growth that will break down organic matter (grass clippings and such) that will help to loosen the clay.
As far as watering goes, I'd only water every two to three days buy make sure it's a heavy watering. Watering less frequent but heavier each time will help the roots of the lawn penetrate deeper and will make it more hardy.
This is pretty much based on my own experience and I'm no expert, so take it for what it's worth! --
"I've learned that depression is merely anger without enthusiasm."