To your points, I'd say
- it's unfair saying it's not a proper convertible. I can confirm this one doesn't shake like a leaf like quite a few convertibles do.
- The engine noise of the V6 is in fact quite nice. Doesn't sound like a diesel. More a heavy, metallic V6 sound.
- Fumes ?
Ever heard of cat's and Euro IV compliance ?
Main drawbacks would be the image (you sure look like a gay dentist in an Audi Convertible) and the reliability long term of the engine (combined with astronomical costs of repairs).
If she likes the car, why not take a 1.8T or a 2.4 V6 petrol ?
I'd definitely do the numbers once and compare how long you need to drive the diesel before you get the premium you paid back.
What's your annual mileage ?