If you want to save at least some of the grass in your run area, you can make frames of 1x4 or 2x4 painted or cedar/redwood/etc. and staple 1/2” hardware cloth over the top. Set the frames wire side up inside the run. Scatter them around wherever you want to leave a patch of grass. The grass will grow up through the hardware cloth mesh. It’s a small enough mesh that the chickens’ feet won’t slip through when they walk on it, but grasses and forbs can still grow through for the birds to eat.
I’d go ahead and set up a couple of ground-contact compost piles wherever you want to in there. Wait, if you like, until they’ve killed the grass. The only way to keep your birds from killing the grass is to give them the whole yard or move them around from day to day, and only have the run for snow season.
We have a lot of aerial predators. As of yet I haven’t had a problem though I’m sure it’s just a matter of time. I like having them out on grass. I don’t know what your situation is, but if you can let the chickens spend most of their time in a much wider area and just use the run on occasion, you may be able to keep it nice and grassy, at least for a longer time. 
People actually use their chickens to destroy vegetation in prospective garden spots. If you have any projects of that sort, the girls will be delighted to assist. Just get some poultry netting, set them up inside it for the day and they’ll go to work. It’s not as fast as a tiller, but it’s a lot less work for you. At night you can put them in the coop for safety and a clean roosting spot, then back to “work” in the morning.