Does it make any sense to grate the radish on a grater without peeling its skin?
Daikon peel is edible. I believe it does have a somewhat different flavor (stronger? I don't have one around to try) than the rest of the root, so you might want to try a bit before you include the peel in a dish. You'll also have to wash a bit more carefully to make sure you don't leave any dirt on. And the skin is tougher, but if you're grating perpendicular to it, you'll end up with small enough pieces that it shouldn't be a problem.