It's fair to say that if they can move mars at all, moving an asteroid of any size is going to be easier.
So broadly, yes. Throwing an asteroid big enough to splat earth like a sledgehammer on a biscuit is by definition possible if you can complete the rest of the plan.
As another answer commented, Earth has taken a rock the size of mars in the distant past and since stabilised its orbit enough we can barely tell. I'm not completely au-fait with the mathematics involved, but what you really want is to smack Earth with an asteroid travelling at a significant fraction of the speed of light.
This will blast earth to shrapnel and then as the sole stewards of the solar system you can clear the debris and move Mars to the better position in the solar system at your leisure.
Of course, if you can accelerate anything to useful proportions of the speed of light, moving mars is more or less an afternoon's work.
Good luck hiding your involvement though. It's hard to imagine that the forces able to impart that much velocity on anything would be subtle.
As some related reading, I refer you to This Treatise on Planet-Busting
Here's an extract from the preamble:
The Earth is built to last. It is a 4,550,000,000-year-old, 5,973,600,000,000,000,000,000-tonne ball of iron. It has taken more devastating asteroid hits in its lifetime than you've had hot dinners, and lo, it still orbits merrily. So my first piece of advice to you, dear would-be Earth-destroyer, is: do NOT think this will be easy.