Sensory Integration problems would be more enduring - i.e., you wouldn't notice them, generally speaking, only during times of change. They would perist.
The most critical ingredient in addressing classroom behavior is the behavior management, including discipline and limit setting. Your son should be placed in time out in the school when he does not comply with the teacher's directions. Obviously, you should insist on his cooperation, and effectively manage his behavior at home as well, but home-based interventions with such a young child are not very effective in addressing classroom behavior. That behavior must be managed on the spot.
I have a 4 year old that is in a full day preschool program. He is in a Montessori school and they do not usually place children in full day until they are 4 1/2. He turns 4 1/2 at the end of November, but he is an exteremly bright child and last spring his teacher and I felt he was ready for full day although he would be the youngest. We are experiencing the same issues you are - he doesn't always listen, has a lot of energy, wants to talk to his classmates and although, does hear the teacher - he chooses to ignore her. Like you, I do not want to classify him as Hyperactive, etc. As I said, he is very bright - he is reading, can write in both cursive and print, can add and subtract and speaks Spanish. I think that he acts up because he is bored at school - that they are not challenging him. They do not want him to read at school, they don't want him writing - just tracing and tell him that he can't do some of the other things he does because he is not an "older" child. Could your child be going through the same thing? I find at home and when I have discussed this with his teacher - that if we challenge him and continue to allow him to learn at his pace - that he is a happier, more well behaved child. Maybe you should discuss that with your son's teacher.