Height of the fence to the neighbor and windows in the boundary construction

That is an annoying situation.

I imagine how one can build a career with the "I hear every word." You only need a recording device now and then and get the material dictated for free. Enrich it a bit and voilà, book, audiobook, and stage program are done. Maybe some cabaret artists have started that way.
Do you otherwise have a good neighborhood? Do you ever meet for some cheerful occasion? An "I hear every word" performance can cause cheerful uproar there.
After that, it either becomes quiet or there is new material. That will then be processed in "Es klingelt um halb 12."

It may not make up for the stupid situation—but seeing comedy in the counterpart is a method to better tolerate unpleasant people. I do that with annoying business partners and sometimes have to slip away laughing after a meeting with such people.
Try this form of reframing.

I will try that next time as well.

I can't say how it is with you but with us it is such that a privacy fence is automatically approved by the authority but the affected neighbor must agree to it. If it is the same with you, the fence probably won't work. Maybe a hedge would work that you plant far enough from the boundary and then can let grow more than 2 meters high. And as an enclosure a normal chain-link fence.

would call the police if she freaks out again

PERIOD

Yes, why? Because she’s screaming? I would have long ago issued a ban on her entering the house. Then the police would get involved. And hey, I would have fun with her. I would sit on the terrace more often in the summer with music at an adjusted volume until 10 p.m. Just as an example. Let her scream until she faints.

What a pity! Simply a pity! You have now built such a beautiful house and now have such a nuisance with the neighbor. In general, it is difficult to give advice on the overall relationship from a distance, but regarding the privacy screen, I also advise: a) either build exactly only what is permitted b) if it violates the usual regulations but is done in coordination with the neighbor, only carry this out after written confirmation from them.