If you haven't had a conference with the teacher, call and make an appointment to do so. If your son's school has a counselor, talk with that person too. If neither of those conferences are helpful, make an appointment to talk with the principal.
Make a list of questions about your concerns. Have some examples in mind. Agree that he's a handful and ask what you can do to help. To start with assume that the teacher is being fair. Always assume she is doing the best that she can do. In other words don't accuse or say anything that sounds like this is her fault. Be as open minded as possible.
My granddaughter is now 9 and in the 4th grade and just last spring diagnosed with ADHD. Every teacher that she's had has told her mother that she talks too much in class, fails to focus or complete assignments, that sort of thing. My granddaughter and her mother have felt that the teachers were trying to help her in this situation. My daughter has liked all of her teachers, some more than others, and did not feel put down by them. Perhaps your son and the teacher do have a personality conflict. Or perhaps this teacher isn't able to be diplomatic.
Does she really use the word bad or does she describe a behavior which you judge to be bad? Needing reminders is not bad. It just indicates that he has difficulty focusing and is perhaps not responding to the reminder by returning to his work.
I would pay attention when he says, "no one likes me in school. All my friends say I'm not cool." This indicates to me that he is having difficulty with his peers as well as his teacher. Perhaps your son reacts negatively to his teacher's comments which then gains his peers disapproval.
I'm remembering the sort of student who clowns and makes "smart" remarks in class, thus disrupting everyone's work and pushing the teacher to react in frustration. You son would probably not do that when you're present. The student who disrupts in class is usually one who feels badly about himself. You could perhaps help him by helping him with feeling success by providing other activities.
Does he have friends with whom he has play dates? It might help if you arranged play dates during which you could observe how he gets along with his friends. Another thing that you could do is to observe him on the playground so that he can not see you. He may just need help in learning social skills so that he's not always pushing himself forward.
Perhaps he feels that he's in competition with his twin sister. She's "good" so he'll be "bad." Comparing them in front of him is apt to encourage him to continue acting up. Perhaps he hears this comparison in school. It's easy for twins to feel competitive. He may feel, unconsciously, that he can't be as good as her and thus go the other direction.
You don't say how you respond to your son in regards to the notes and his venting. Your response is an important part of his reaction. It's a good idea to support the teacher while being empathetic with your son. That is tricky, I know.
You say that while you're visiting the class you're there "to make sure he does what he's supposed to be doing." I understand the feeling that you're responsible for his behavior but that feeling is not helpful. Try just observing. Sit in the back of the room in a place so that your son cannot see you. Do not make comments. Be as invisible as possible.
It seems that many parents feel that they are responsible for their child's behavior and that is just not true. They are responsible for teaching their children how to behave but once the child goes to school they become responsible for their behavior. When the parent sees that their child is behaving in a way that they do not approve of they are then responsible for teaching their child the "correct" behavior. So please don't feel responsible. Be a neutral observer as much as possible.
The teacher may be saying something to your son and not the other children because she wants you to be aware of the behavior. Try to focus on the interaction between the teacher and your son, especially on how your son reacts to her comments. Also try to see if her comments may be an attempt at managing with humor. The comment about when you're absent we have peace and quiet can be said with humor. I've heard that said to my granddaughter and everyone, including my granddaughter laughed. She's aware that she talks all the time and that she should be quieter.
And there is hope. She is quieter and more focused now. I think the teacher she has this year has helped that progression. Her teacher is also high energy and a talker. She knows how to direct high energy kids.
If you can be calm and portray a sense of partnership with school people they will work with you to find a way to help your son be a better student.
I don't know what you mean by a behavioral health doctor. If he is someone who evaluates for ADHD did he do that? My granddaughter is now taking medication and that is a part of her improvement. Her mother refused to try that for a couple of years but finally gave in to see if it would help and it has.
I'm glad he's having success with his tutoring teachers. It might be helpful to talk with them about how his difficulties in the classroom and see if they have any suggestions. Do remember that working with a tutor one on one is altogether different than working in a classroom.