The moon is 700 times farther away than the starlink satellites (or twice that, if you consider the bounce), so I find it hard to imagine that it would be impossible to communicate with much closer satellites over laser when both sides can have an active transmitter.
However, this solution is going to stop working when a cloud drifts past.
Not really, because you'd be using a frequency that passes through clouds. A snow storm or hail is impenetrable, and there are weather events that cause a 1-2 second blackout, as well as cause refraction (which is mostly a challenge in reaiming the beam fast enough to compensate), but anything in the air is fine. Clouds, mist, ... But is aiming at a 1 arcsecond target moving across the sky at at least 1 degree per second from a normal (ie. moving) building really doable with "standard hobbyist telescope mounts" ?
I know 5 years ago we were still doing this with lasers on rockets toward planes, because planes can just keep their angle to a rocket essentially constant. I know there's experiments doing direct laser to satellite, no idea how well that works.
The clouds are however much more of a problem than you're suggesting. One promising infrared band is around 10 microns, but a thick cloud will still scatter that. You'd need a 20cm wide laser beam at that wavelength for it to diverge to a beam width of around 10 arcseconds. Which is basically a reasonably-sized telescope, working in reverse.
Alternatively, you could go for millimeter waves, which would pass through the clouds reasonably well, but then you're well outside the realms of "laser" and into the standard directional dish antenna. And it'd have to be a very large dish to give you a narrow beam. For instance, a rather unsubtle 2 metre wide dish with a 1mm wavelength will give a beam that diverges by 100 arcseconds. And there will probably be omnidirectional leakage which the dastardly authorities are likely to be able to detect. At least visible and infra-red leakage can be easily blocked and concealed, but radio is much harder.
The only thing that could bypass is GPS + laser links (meaning physically aiming a laser both on the ground AND on a satellite). You cannot detect that without being in the direct path of the laser (though of course you can still see the equipment aiming the laser, so it doesn't just need to work it needs to be properly disguised). That requires coherent beams (not easy, but well studied), aimed to within 2 wavelengths of distance at 160km (so your direction needs to be accurate to 2 billionths of a degree, obviously you'll need stabilization), at a moving target, using camouflaged equipment.
This is not truly beyond current technology, but you can be pretty confident even the military doesn't have this yet.