In my first response, I assumed that there was still some life in the trees above. Otherwise, as I have already written, they should be removed.
It is normal for trees to sway in the wind, especially at this height.
I would never remove branches from foreign plants, even if the owner agrees. Because if the plant dies afterwards, the person who cut it might be blamed by the owner, even if the cutting had nothing to do with it.
Furthermore, it should be considered that if you remove branches from only one side of the trees, there will naturally be a higher weight on the opposite side, which can cause such a tree to fall in the direction of the heavier side. Thus, in the direction of the buildings. I would not rule out that the person who removed the branches might also get more or less a lot of trouble.
Therefore, I would always let the owner do this.