The round storage "boxes" I have cannot fit the turret as your pictures show at all. There is a 1" diameter tube in the way (in mine, purchased in 2010). The inside diameter of the tube is just the right size to fit the sizing die (barely) or the case-mouth belling (easier), so putting the turret (with dies installed) into the round box UPSIDE DOWN seems like the intended fit. I drop the shell holder in first.
The top of the die box goes on without any trouble at all.
If Lee does not have a solution (which would surprise me if they didn't), a 4" diameter section of PVC plumbing (drain) pipe should do, as the turret is just a little bit over 3" in diameter.