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)?
I think it's better to hold the note. Using a sustain pedal would also sustain the notes in the bass as well as the upper treble notes. It's difficult to sustain only the single note, even using the sostenuto pedal (which would normally sustain the first D chord in measure 9.)
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.