I would ask the pediatician before you see a psychologist. He is able to leave the family and do other activities, I would keep explaining that you all are fine and he doesnt need to worry.
Your question - if this has been going on now for more than twelve months, should I have him see a psychologist - My answer - yes. It appears the anxiety your son is experiencing is interfering with his life - that is the "clue" that your son requires extra help. By the way, a psychologist usually works with the parents and offers advice, and ideas/hints to the parents to better help their child. Since each child is different, it usually takes a professional to guide parents through this type of "intervention".
I suspect your son is suffering from GAD (Generalized Anxiety Disorder) or SAD (Separation Anxiety Disorder or perhaps - a combination of both. I might suggest you google the term "childhood anxiety" or "anxiety disorders in children" or similar words/phrases to become more educated about this very common disorder (it is estimated that approximately 10-13% of the population suffer from some form of anxiety). Then, it is time to see your son's pediatrician or family doctor for assistance. If he/she is unable to help you, ask for a referral to a medical person with experience in anxiety issues.
The reason your son is able to interact with his peers is because he is not "afraid" for them or of them. This type of anxiety would be called "social phobia" and your son does not display any of the behaviours. This is common for children who suffer from GAD and sometimes (although more rare) SAD.
If anxiety is the issue, then your son will not outgrow it nor will it go away (but you already have that figured out). Since your son's teacher has commented on his behaviour as being "not the norm", then it is time to seek advice and help. I wish you the best ....
Has he heard any worrysome talk at home regarding the safety of his Family?has anyone left or died and he is concerned the same will happen to you, how about his friends has there been an upset there he could have witnessed or heard about? anything scary on the TV he seems overly protective for a young child. Check out all avenues before getting further help.