Approval for basement office - Freelancer (digital work only)

The letter from the building authority also points out "since the development plan designates a general residential area, the exceptions according to § 4 paragraph 3 of the Federal Land Utilization Ordinance are excluded."

So, I think something is being misread or misinterpreted by you.
If you are purely working at a desk, without customer traffic etc., then you are not listed in paragraph 3 at all and do not need special approval. You also do not have to be listed. Paragraph 3 does not concern you at all.

After all, it is not a pure residential area, but a general residential area.

Your case is found right in the first sentence, namely in the word "mainly." And there is no reason not to approve desk jobs. If there is currently a busy newly qualified person at the building authority, in my opinion, she will earn a disciplinary action.

I thought so too at first. Unfortunately, it’s not that simple. Please google "Etwas anders sieht die Situation aus, wenn du in einem allgemeinen Wohngebiet wohnst." The first website in the results has a good explanation.

What is strange... On the website it says "Does this also apply to freelancers and solo self-employed?" But only freelancers are addressed below. However, I am solo self-employed and need a business registration for this.

I thought so too at first. Unfortunately, it’s not that simple. Please google "Etwas anders sieht die Situation aus, wenn du in einem allgemeinen Wohngebiet wohnst." The first website in the results has a good explanation.

Why are you reading an advertising page of a rental company from Ahrensburg (SH) if you come from Bavaria? That does not give you legal certainty.

Nevertheless, the building authority now writes to me:
"The commercial use of the office in the basement is not possible, as the development plan designates a general residential area, but excludes exceptions according to § 4 para. 3 of the Land Use Ordinance. Therefore, we cannot grant an exception for use as a non-disruptive business. Accordingly, only a home office would be possible."

Ah, missed something. They have already rejected it.
So, I would sue against it.

The basic rule (the municipality decides whether an exception is possible) seems to apply to all of Germany. I will still contact a lawyer. However, this does not look very promising for my situation. Yes, exactly, already rejected for that very reason.