I don't have too much advice on making the choice, since we have gone the private school route all the way. And our daughter is not AS, although she is HG and has some social issues. But if I were you I would ask the psychologist for more information on why he is concerned about the public school option. He may know some specifics (for example, that the administration is not supportive when kids are bullied, etc.) that he has not shared with you yet.
Since your son's issues are primarily social and that is where his friends are going, that could be a bonus for him. It would be much harder to enter a large public school where he didn't know anyone, but if those friendships are relatively strong it could be a big help.
Here is another question: what services have you been getting in his current school environment? That might help you determine what you are may need going forward. For example, our private school has 1.5 FTE school psychologists that have been quite useful from time to time, along with a learning specialist. We wouldn't get that same level of support in the local public high school.
One thing to keep in mind is that this decision does not have to be permanent. If he tries the public school and ninth grade doesn't go so well, you probably have the option to switch him to the private school.
A final comment, not sure how you sit regarding college funds, but I have one in college and one in a private high school right now. I would advise you to give some hard thought to the finances involved. If you have a good public option, you can save a bundle of money by using it IF it works for your kid. I'm not saying you should in any way sacrifice your son's happiness & well being at this level if it comes down to that. But if all things look pretty equal, it might be worthwhile to give the public high school a try. If your son is that bright, he may have some great college options, but I saw several of my kids' high school friends turn down some great college opportunities because of finances. If you are financially "in between" and your son is not in an underrepresented group, it can be tough to wrangle much financial aid even for a bright kid.
