Depends quite a bit on the wavelength of the laser beam. Not to mention, these things are mega-watt range. I doubt you could get anything through the atmosphere without picking up something. I work with industrial lasers every day, a fleck of dust on a lens is enough to cause a burn spot and wreck the lens. Scale that up to Mega-watt ranges and even water condensate on the surface of the missile has probably enough absorbtion to make the polish unimportant. We're talking about beam spots several feet across with mega-watts of energy. I know the mirrors in the lasers I deal with are Molbdnenum, tough as hell, but I suspect way too heavy and impossible to clad a missile with it. Anything else and the laser is just going to burn through it.
I suspect this has been taken into account by these guys, which is why it's SO powerful. Even if only a few percent gets absorbed, it's prolly way more that it would need.
Wouldn't this laser have to be so accurate that it only fires when it knows it will hit the target and immediately stop once the target is destroyed? If not, wouldn't this beam go on to destroy an unintended target?