No you shouldn't give up.
Can you do private testing? We had similar issues and concerns about behavior and emotional responses. We did private testing and got our son into first grade, skipping K. He is now in 2nd but in 3rd for math. It was the best thing we could have done for him. There are really no more behavior or emotional issues, other than "typical" OEs! It was a fight worth fighting because now the principal is on our side. He's the one that tested for the second acceleration in math. And our son isn't as advanced as yours sounds. I'd place mine at level 3.
