As for playing B Ionian over A major, a car crash is coming up. You might manage to pull it off if you play B mixolydian, but the 7th in B Ionian is A#, which would mean you would have A as the root note of the chord, A# as the 7th, and B as the scale resolution note. Ouch. To get the jazz sound, play the min7 (like Christoph ssaid), but then move to, say, the domV chord, so it sounds jazzy, rather than moving to another minor.