Is sound insulation according to DIN standard 4109 relevant for building permits?

Apologies first and foremost that we are laypersons and ask questions.

One does not need to and cannot apologize for being a layperson. Laymanship only becomes dangerous when it is based on dangerous half-truths (DIN 4109 does not apply "to us" at all, "is difficult to achieve") or complete ignorance like user Payday.
By the way, broadly accusing the gentlemen of the building supervisory authorities here of incompetence fits perfectly into the picture...

We have now had a paid expert report created, which has shown that our sound insulation measures lead to significantly better values than required.

There you go. The mid three-figure amount that a properly conducted sound insulation certificate against external noise costs is gladly invested with respect to planning and future security. Especially then

“DIN standards are not legal norms but private technical regulations with a recommendatory character.” (BGH, Case No.: VII ZR 184/97)

They are only binding if the legislator explicitly prescribes them. Otherwise, compliance with the DIN must be contractually agreed upon. However, the executing company may then have to prove that the work meets the "usual professional requirements." Since things often get spun around in circles here, this is not exactly the same, although it would theoretically come close. We have had several such cases professionally ourselves. In none of these cases did the customer succeed in the end. The DIN is usually also very vague, so that you can devise the solution yourself.

„DIN standards are not legal norms, but private technical regulations with a recommendatory character.“ (BGH, Case No.: VII ZR 184/97)

No one has claimed otherwise. The question remains why it is repeatedly pointed out.

They are only binding when prescribed as mandatory by the legislator.

Take a look at your state building code. Usually §3, paragraph 3. It is actually unnecessary to have to point out the state building code in such a forum.

DIN 4109 has been introduced throughout Germany as aRdT ("functional requirement," "execution specification," or "instruction") and is therefore more than a private technical regulation with recommendatory character.

So then we’ll get back to you once more. First of all, apologies that we are laymen and ask questions. I thought that’s what forums are for, anyway.

We have now had a paid expert report prepared, which showed that our soundproofing measures lead to significantly better values than required.

The report was submitted and the approval granted.

Dear Turks84,

I really hope you read this soon and can help us further. We are facing exactly the same problem as you described in the initial post.

My question now is, where did you have the report prepared, how fast was it done, and what did it cost? Feel free to contact me by private email or phone. I would send you the phone number in a private message.

We are more than desperate. Our entire construction project seems to be failing because of a few dB.

Many thanks!

We are more than desperate. Our entire construction project seems to be failing because of a few dB.

Keep calm, no reason to despair, even in your case the construction project will not fail due to the lesser requirements of the 4109.

My question now would be, where did you have the expert report prepared, how fast was it, and what did it cost?

I would not recommend getting anyone from anywhere. Call your building permit authority or the local emissions control authority and ask for contacts. The usual local offices know what they have to deliver to the specific authority, and the authority knows what it will get. Such a sound insulation verification according to 4109 should not cost more than 1.5 T€ in your area either; the workload is at most 2 days, as this is a relatively simple number crunching in Excel.