Let's say you break this puppy into dozens of large pieces. They will indeed spread out around the original trajectory, by many thousands of miles over a decade in all directions. As it is, the odds of the unified asteroid hitting the earth is extremely unlikely, the earth is small and empty space is huge. With dozens of large pieces spreading out in a pattern around the original trajectory, though, the odds of one or more intersecting earth's orbit go way up, and each one could do a whole lot of damage. Trying to destroy several smaller pieces would be a lot more complicated than dealing with one large piece, which is difficult enough, but at least it is easy to find and track.
Fooling with this until we are dead certain where it is going and what the effects of our meddling would be is not a good idea.
I wouldn't mess with it until I figured out which continent or ocean it would hit. If it would impact China, for example, they should pay the bulk of the cost of diverting it.
If it was going to hit Massachussetts, we'd do nothing at all.