I think the less equipment the better. Also try to do exercises that strengthen his top line so he becomes stronger and can keep his head still, but you also need to try to keep your handle as still as possible. It’s probably a hard concept for him to grasp but he’s attempting to stabilize himself with you on him it’s harder if you can’t keep your hands still!
Do not worry about the head yet if he is green. The head will relax when he relaxes and learns to do his new job. Not the answer you want hear I am sure, there are things to do that could prevent the habit but those are only band-aids and crutches that if you remove them you have the same problem. Time is what it takes and LOTS of patience