A short update:

andrewclaus
I would take it apart, as shown here.
-> I have the newer version, which, as far as I can tell from my research, cannot really be disassembled.

Iride01
Most wire will be too bendy.
-> Yeah, I tried with some steel wire first and then went to a dentist who gave me a hook and one of those scratchy right-angled things.

Thanks all for the suggestions. I tried everything and had two friends try it as well. However, none did manage to get the wire out, so we ended up drilling out the housing. As you can see in the photo, a slit goes up from the "inserting whole" for the wire. Not sure what the whole/slit is for exactly, but we drilled the whole out with a 4.8mm drill so we could just push the cable terminus through.
It all seems fine, but I'll try tomorrow with a new cable. I already CAD designed an insert for the hole to get it back to its original size - might 3d print that tomorrow as well and glue it in.
It was quite easy to drill it out and I do not think we damaged anything.

There is a new photo in the album where you can see the hole we drilled -still cannot post the URL

Thanks again to all and I will keep you posted.