The atmosphere is hardly a problem. You'd use numerous beams simultaneously, fired from all over the country. No single beam would be powerful enough to cause atmospheric breakdown. The sensible choice is to surround every power plant with lasers.

No, this won't be undetectable.

Light pressure is fine, but probably not as productive as ionizing the surface to produce thrust. With high power, atoms at the surface can become multiply ionized. They get blasted off the surface. This would be pulsed, since otherwise the resulting ions would absorb some of the beam and there would be a risk of melting the surface.

If the thrust isn't enough, for example due to a very high orbit, you can just keep going until the target is gone.

Weather isn't much of a problem. There is probably clear weather somewhere over the country. Station keeping won't last forever.

Oh, and if you don't insist on a neat and tidy removal, you can just use non-pulsed beams to melt the target.