Ok, W, as a difficult child, my two cents about evaluation.
It is right you cannot force him to get an evaluation unless he is an immediate threat.
But there are ways to do to convince him and ways to fail.
The most important, and if you had to retain only one thing, it is this one.
He does it not for you, not for his dad, not for anyone else, he does it for himself.
Never ever bring the request like it is for you. It will not work at all, it indeed will be the best best way to make him refuse and he won't change his mind.
Assure him that the evaluation won't change that you love him as his mom, and whatever the outcome, he is still your son you love unconditionnally. For you, it is obvious you love him whatever the result of this evaluation, but it may not be as obvious for him.
He will probably bring the concern of his friends. You can tel him that he has no obligation to tell his friends, but if his friends are really his friends, they will accept it like he accepts his blue/green/hazel eyes.
That he is not alone to have such issues.
Don't threaten, don't drag him.
Tell him how much you are concerned because you really love him.
Reassure him that he can tell if/when he does not feel at ease with someone, even if they are doctors/psychologists....
It may take some time, but it will bring the best results.
You don't attract flies with vinegar, you attract them with honey.
Assure him that whatever his issue, he is still a person who deserves help and happiness. And that he needs help to be happy, like some persons need help because they don't know how to cook/paint...
