R.R.
Have you asked him why he doesn't want to do it? Maybe you could alleviate his fears if you know what they are. It could be the closed room, maybe they can crack the door a bit, or he's afraid of the equipment and you can explain what it's for, etc.
Or what about acting as if he doesn't need to do it, tell him "Mommy needs to do the eye chart exam," but take him in with you, sit there yourself and do it, but say the wrong letters to see if he corrects you?