You can't just expect to slow the Moon down by putting an asteroid or similar sized rock behind it. Our Moon is 5th largest Moon in the solar system if I remember correctly... and it's much larger then most asteroids, including both Mars' moons.
If you wanted to lower the Moon's orbit, you'd have to slam Phobos into it with quite a significant velocity... but even that would produce an elliptical orbit, which could bring up a hell of a lot of problems for Earth.
And to answer your question about VASMIR: No. No rocket engine, no propulsion system... nothing is anywhere near powerful enough to "move" the Moon, or Phobos out of Mars's orbit, for that matter. Phobos is significantly lighter... but still 10^16 kg.
And before someone asks about nuclear weapons, I doubt that'd work either. The energy from the reaction has to go somewhere, but since there is no air or atmosphere, you can't expect a blast wave to "push" the Moon.