What about the opposite - how feasible is it to "control" a Lagrange point even if you wanted to exclude other equipment?
At minimum, you need weapons coverage over a complete sphere about the point you want to protect. If your claim is larger than your spacecraft, you need multiple in a 3D perimeter.
But even supposing you had enough weapons, it would take much more effort to destroy all the shrapnel before it hits you.
And even then, if someone decides to throw something very heavy at you very fast, there's sweet FA you can do about it. If I throw a big satellite or a ship into a kamikaze vector at you, your one and only option is to move out of the way as fast as you can. Even if you hit it with all your weapons at once, you still have almost as much mass flying at you almost as fast, but now it's buckshot instead of a cannon ball.
It really isn't practical to defend any amount of space, not even a planet.
Controlling a Lagrange point would have to be a political mechanism. You can't really physically enforce it in space. You'd end up fighting wars on the ground over a point in space.
If the Lagrange point is indeed unstable the random movement should be rather efficient to achieve. And kinetic weapons would need luck to hit you. For explosive weapons you might be able to rely on the gatling guns.
It seems to me like the Lagrange points might be one of the few things in space you could occupy and deny an enemy. Perhaps LEO should added to that because it can be hit from the ground.