Update:
Since my car still has a worldwide warranty and I naturally wanted to handle it through that, I went straight to the Volkswagen service center here in Graz on Wednesday.
Okay, I accepted the offer, explained the problem, and left the car there. Then, just to be safe and to inform him beforehand, I contacted the salesperson who connected me with the car.
The car was ready on Thursday... the sensor was replaced, the DPF was regenerated... AND:
MSTG updated to software version 4356!
I've already driven about 300 kilometers, but I haven't been able to notice any real differences yet, neither in fuel consumption nor in top speed. I need to drive a bit further to see if there are any changes.
Regarding acceleration, I can say that it sometimes feels a little "gentler." It feels like the turbo boost builds up slightly later/slower, especially during kick-down situations on the highway. There was even one instance where it jerked noticeably when I shifted from 6th to 4th using kick-down. But so far, that's all I've noticed!
He's likely very close to achieving top performance... I would be happy to create some logs if anyone is considering doing such an update or simply wants to know where the actual differences lie.
The shifting problem has largely subsided, although it still seems a little slower to me (probably very subjectively) than it was before the exhaust system warning light came on.
Here's what I have to offer... if anyone wants to compare logs, I'm happy to help - if there is a need for it

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Thank you, and best regards from Austria.
Alex.