I'm not deliberately on the Ornish diet but I have an intolerance to saturated fats so have a fat intake similar to Ornish levels. For me, eating saturated fats hurts so I quickly learn what I can and can't eat.
I'm a very new cheesemaker so can't speak to making cheese from nonfat milk, but I can tell you what works for me in terms of consumption. Most commercial cheeses are out for me, but I do find I can eat a little parmesan because of its lower fat content. I've also had success with home-made feta and halloumi made from low-fat milk - but they will, of course, increase salt intake if that's an issue.