First the bad news...I can hear the keys to prominently too. Not sure sax recording technique, but I'll bet if moving to the side doesn't work, you could rig some sort of transparent shield like a piece of plexiglass to shield the key noise. Or maybe oil them?
Dude, I've heard some shit and you are AWESOME at your instrument. Holy SHit, man. Careless Whisper was a nice throw-in, too. It is TRULY refreshing to hear someone so in touch with their music, and their instrument. Just get the key thing resolved for recording and you are set.
My hat is off to you, sir.