Store bought grated cheese has a coating on it to keep the graded pieces from sticking together. You’re better off buying a block of cheese freezing it and then grading it. Freezing it makes it a lot easier to manage and you can also use a food processor to grate it if cheese is Frozen.
Freezing the cheese sounds like an interesting trick although I don’t find it all that hard to grate even when it’s room temp. What does the freezing do that helps?