Energy equals momentum. Niven and Pournelle in assorted novels (i.e. The Mote in God’s Eye) suggested that a decent sized very light-weight sail spanning dozens or hundreds of kilometers could use the momentum from reflecting light from the sun or giant lasers to reach decently relativistic speeds,
The obvious solution to an orbital laser would be … wait for it… orbit. During half the orbit it blasts away at the target and is pushed faster in its orbit around earth. then as it comes around from behind the earth and starts blasting again, the same amount of momentum slows the laser down as much as it was sped up. the orbit just has to be in the same plane as the beam direction. Just don’t blast away while in front of the earth, or you will progressively push the perigee closer to the earth until reentry and rapid unplanned disassembly happens.
Again, the issue will be dissipating the left-over energy from firing the laser, since lasers are inherently inefficient. I think landing a large (LARGE!) number of ion engines and regularly refueling them would be the solution, given enough time. After all, given enough time, the deflection amount does not have to be large.