I really no not do push-up assessments at all. They are for the fit and healthy, and the assessment alone could to more harm than good.
I often have clients stand against a wall and bring arms overhead and out to the side to see range of motion, whether the client can bring the arms against the wall, how much he compensates to make that work (if shoulders are okay).
Balance assessments can also be valuable. Ask the client to stand on one foot and see how long he can hold his balance and observe his balance strategies. I always take a lot of pictures. It’s amazing how patterns emerge when you put the pictures side by side.
Good luck with your client.