All restaurants suggested are top class, the best and very fitting. I would go to Vau because they have no music and you could talk quietly, maybe even sit in the small inner court yard. Reinstoff is more conspicuous and showing off technique. Vau is feel good food.
Just to add to your embarrassment of richness - why not consider Die Spindel as well.
Kinda depends on what your budget is. I think Hartmanns may well be on the same level as Reinstoff, whereas Vau might be a bit "cheaper".
Have you looked into Horvath, ETA Hoffmann, Dos Pallilos, Weinbar Rutz?
Those are all very valid options. Noi Quattro for fabulous Italian, or Tim Raue for (very) expensive Asian-inflected cuisine.