If you dump a mound of compost where you want to grow turf and then dig around in it ONE SEASON later you will see the feeder roots from your oak trees. You can't compete with mature oaks unless you put a barrier between them and your new lawn. Moss works great but does require some maintenance (but not as much as grass).

moss is wonderful- but you do have to keep leaves & such off of it, or it'll die back. Use a blower- raking tears it up. It can handle a decent amount of foot traffic. Keep it moist- it'll brown and go dormant if it gets too dry, but usually once it's moist again it just greens up super fast. There's a guy who just does moss who sells stuff in north hills shopping center. It's called moss and stone gardens. You could look them up and get lots of good advice.