With this game, the "near enough is good enough" way of thinking is the way to go. You will never really get it to match up perfectly. At least not on purpose.

My biggest tip: IGNORE the circle. It's a guide only. The grass/ground will feather out past that.

I would use beams to mark out the perimeter of your grassy area, then carefully put the grass in. Start in the middle and move in very small increments towards the sides to get the grass to butt up to the beams. You can then use the hoe to re-flatten any spots that go over on the other side of the beams but again, you have to move in very small increments to avoid removing the grass you've just carefully placed.