The moon still doesn't feel like a source light. It feels more like it's part of a backdrop painting. A real full moon is always the brightest object in a scene, absent any other lighting sources from the ground that might overpower it. I've taken lots of pictures of full moons before and what I learned was that it's MUCH brighter than you think it is. If you were to stop a real camera lens down to allow the moon to look as it does in your render the rest of your entire scene would be entirely black, so you've got the inverse of what it would look like photographically at this point.
-Greg