I agree about the time element. But if the rockets don't use some of the asteroid for the main reaction mass (and solar power) there's no way to afford lifting enough to make a real difference.
They probably should otherwise it will be VERY expensive (although the alternative is too)
Maybe there's a way to use solar sails even. MIT students suggest using white paint to force asteroids to higher (slower) orbits for safety but to me that's just setting up future problems.
Solar sails will work up to a certain distance in the solar system. Check the site of the planetary society, they are busy with such a thing.
Painting the thing white (for 50% !) will cause the "gas bursts" first on the dark sides giving it direction. There is however a serious "risk" you need to repaint several times.
BTW, "gas bursts" include stone and dust can damage solar sails.