The orbit of the asteroid would have to be known to a LOT of decimal places...a tiny error at the orbit of Jupiter would result in a BIG ERROR on Earth. Of course, this could also be used as a weapon...a really good one, too, because by the time it got close anything you could think of to deter it would be about as effective as an umbrella.
IMHO, it makes much more sense to take small asteroids moving retrograde, attach these long-working motors to them, and keep them handy for threat annihilation. We’d need some large but available number of them (or we could park them at the Lagrange points), and thread the needle with just one. A single head-on impact by a small one would basically obliterate both our projectile and the interloper. Problem solved.