I'm not familiar with the bass editions, but in theory, if you play the solo line down an octave you should play the bass line down also. The composer wrote only those two lines and to play the solo down an octave creates relationships that are not intended and often not proper for the period. Nearly all the chords will be in the wrong inversions. Having said that, the average non-musician modern listener probably won't notice and will hear the music as monodic and not contrapuntal.
Playing the bass line down on the piano can sound muddy. You need a pianist with a light touch. Maybe on the harpsichord it sounds clearer, but I've never heard it done.
"Happiness is not a riddle, when I'm listening to that big bass fiddle." www.thesymphony.org