I agree with suggestions to explore alternative methods of assessment if his teachers will cooperate. Whether he needs a different type of tasks on written tests, whether he needs to be allowed to dictate answers for someone else to record, whether he needs to be allowed to provide strictly oral answers... Depending on what the psychologist and teachers and parents and counselors and whomever else becomes involved determines, it may be necessary to develop an IEP-type of structure that would require adaptations to support special needs that are identified.
"He says that when he finishes the tests he thinks he has done well on them." Has anyone had the opportunity, after he has finished a test, to have him orally go through the test and give verbal answers to all items immediately? This might help to confirm if he does, in fact, understand the material and for some reason isn't able to show that knowledge in the test format.