You are correct in that most "hobbyist telescope mounts" are good for tracking stars at ~1 arcsecond, only where those stars don't move across the sky very quickly (up to 15 arcseconds/second). However, it is quite within the realm of "hobbyist" telescope mounts, albeit towards the upper end, to track orbital objects. I have seen an example of a telescope mount tracking the international space station to get good images, and the tracking was pretty solid. It is assisted by a secondary telescope on the mount that helps the mount maintain good tracking, not just pre-knowledge of where the object will be.
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.