You can probably build the laser at an Earth L5 point. From there it should be easy to keep it pointed correctly, and it would be far enough away to power it with a fairly hefty reactor, without requiring all kinds of shielding. Alternately, by going a bit farther out, it could be stationed at the appropriate pole of a large asteroid, which might make it easier to build, since it could use the asteroid as the “framework”. Such a location could also provide the benefit of a little gravity, making maintenance a bit easier.
“You can probably build the laser at an Earth L5 point.”
And I promise you can trust me with it.
-Emperor Conejo the 1st