In the following measure of Air in D major by Bach, do I have to keep pressing A (and play around with thumb on A) or can I let go (with sustain pedal)?
This would completely mess up the bass pattern, so press A and hold. That's not really much of a stretch: this arrangement of the orchestra suite #3 movement is for typical keyboard instruments. As a note aside, I play this thing from the orchestra score on a chromatic button accordion and for a lack of skills at tasteful reduction of polyphonic music, I just play the violins and the bass, leaving out the viola. That actually requires you to span a 2-octave range at one point of time.
So if you get a piano version of an orchestra piece that seems awkward rather than impossible, chances are that you already have a version that the arranger considered playable as-is.
As a note aside, I remember some piano menuet by Mozart apparently requiring the use of the nose for a note in the middle with other keyboard action left and right.