What a wonderful teacher you must be to care so much!
Have you discussed with him that it's noticeable? If he wants to succeed that badly, he might figure out a way in his own mind how to calm that down? I know that when my anxieties were so high that others would comment, but yet I felt I had no control, I would figure out a way to get things under control...even as a child. It's strange, but I could always do it when someone commented. It was a valuable tool to practice when I later began to sing publicly.
So possibly if you kindly mention this, he might know already inside what he needs to do? If not, you could mention a breathing technique, or going to a safe/fun place in mind before playing? I knew that a stiff, frightened looking singer would take away from the total performance...greatly.
Possibly you could get him really comfortable in talk, laugh, etc., then just when you see his body at the most relaxed point it out to him and ask him to recognize his body right at that moment...recognize the feelings, the muscles relaxed, etc. Then ask him how he thinks he might keep that same feeling while he's going to his instrument. He might be able to figure out a way to do just that.
I hope you keep us posted on this. What a treasure you must be as a teacher.
KD